Implicit explicit methods forhyperbolic systems with stiff

sourcesLorenzo Pareschi

http://utenti.unife.it/lorenzo.pareschi/

Department of Mathematics

University of Ferrara, Italy

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 1/42

Outline

Hyperbolic problems with stiff sources

Space discretizations

Finite volumes

Finite differences

Time discretizations

IMEX Runge-Kutta schemes

Applications

Hybrid methods

Hybrid Monte Carlo (HMC) methods

Applications

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 2/42

Outline

Hyperbolic problems with stiff sources

Space discretizations

Finite volumes

Finite differences

Time discretizations

IMEX Runge-Kutta schemes

Applications

Hybrid methods

Hybrid Monte Carlo (HMC) methods

Applications

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 2/42

Outline

Hyperbolic problems with stiff sources

Space discretizations

Finite volumes

Finite differences

Time discretizations

IMEX Runge-Kutta schemes

Applications

Hybrid methods

Hybrid Monte Carlo (HMC) methods

Applications

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 2/42

Outline

Hyperbolic problems with stiff sources

Space discretizations

Finite volumes

Finite differences

Time discretizations

IMEX Runge-Kutta schemes

Applications

Hybrid methods

Hybrid Monte Carlo (HMC) methods

Applications

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 2/42

Hyperbolic problems with relaxation

Many applications involve hyperbolic balance laws withsource terms of the form

∂tU + ∂xF (U) =1

εG(U), x ∈ R,

where U = U(x, t) ∈ RN , F : RN → RN and ε > 0 is calledrelaxation parameter. Several kinetic equations have thesame structure with U = U(x, v, t) ≥ 0, v ∈ R, F (U) = vU .

The numerical solution of such systems may bechallenging when the source terms are stiff and it isnumerically expensive to fully resolve the small time scales.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 3/42

Hyperbolic problems with relaxation

Many applications involve hyperbolic balance laws withsource terms of the form

∂tU + ∂xF (U) =1

εG(U), x ∈ R,

where U = U(x, t) ∈ RN , F : RN → RN and ε > 0 is calledrelaxation parameter. Several kinetic equations have thesame structure with U = U(x, v, t) ≥ 0, v ∈ R, F (U) = vU .The numerical solution of such systems may bechallenging when the source terms are stiff and it isnumerically expensive to fully resolve the small time scales.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 3/42

Examples & related problems

Discrete kinetic models

Gas dynamics (gas with chemical reactions, Extended Thermodynamics)

Granular gases

Semiconductors models

Traffic flow models

Shallow water equations

...................

Kinetic equations (BGK, Boltzmann-like)

Convection-diffusion-reaction equations

In this lecture we will survey several numerical methodsthat are designed to capture the macroscopic physicalbehavior without resolving the small time or spatial scales.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 4/42

Examples & related problems

Discrete kinetic models

Gas dynamics (gas with chemical reactions, Extended Thermodynamics)

Granular gases

Semiconductors models

Traffic flow models

Shallow water equations

...................

Kinetic equations (BGK, Boltzmann-like)

Convection-diffusion-reaction equations

In this lecture we will survey several numerical methodsthat are designed to capture the macroscopic physicalbehavior without resolving the small time or spatial scales.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 4/42

Numerical aspects

Difficulties

Nonlinearity of the flux function. If F (U) is nonlinear, the solution may develop ajump discontinuity in finite time.

Stiffness of the source for small ε. Some implicitness is required in the treatmentof the source.

Computational complexity. The dimension of the system can be very high (typicalof kinetic equations).

Requirements

The schemes should be high resolution shock capturing, yielding correct shocklocation and speed without numerical oscillations.

It is desirable to develop schemes which are implicit in G(U) and explicit in F (U)

(IMEX).

For high dimensional systems some degree of stochasticity is required originatingan hybrid method.

Related topics: Well-balanced schemes, Domain decomposition techniques.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 5/42

Numerical aspects

Difficulties

Nonlinearity of the flux function. If F (U) is nonlinear, the solution may develop ajump discontinuity in finite time.

Stiffness of the source for small ε. Some implicitness is required in the treatmentof the source.

Computational complexity. The dimension of the system can be very high (typicalof kinetic equations).

Requirements

The schemes should be high resolution shock capturing, yielding correct shocklocation and speed without numerical oscillations.

It is desirable to develop schemes which are implicit in G(U) and explicit in F (U)

(IMEX).

For high dimensional systems some degree of stochasticity is required originatingan hybrid method.

Related topics: Well-balanced schemes, Domain decomposition techniques.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 5/42

Numerical aspects

Difficulties

Nonlinearity of the flux function. If F (U) is nonlinear, the solution may develop ajump discontinuity in finite time.

Stiffness of the source for small ε. Some implicitness is required in the treatmentof the source.

Computational complexity. The dimension of the system can be very high (typicalof kinetic equations).

Requirements

The schemes should be high resolution shock capturing, yielding correct shocklocation and speed without numerical oscillations.

It is desirable to develop schemes which are implicit in G(U) and explicit in F (U)

(IMEX).

For high dimensional systems some degree of stochasticity is required originatingan hybrid method.

Related topics: Well-balanced schemes, Domain decomposition techniques.Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 5/42

Space discretizations

Consider first the case of the single scalar equation

ut + f(u)x =1

εg(u).

We distinguish between schemes based on cell averages(finite volumes) and point values (finite differences).

Let ∆x and ∆t be the mesh widths. We introduce the grid points

xj = j∆x, xj+1/2 = xj +1

2∆x, j = . . . ,−2,−1, 0, 1, 2, . . .

and use the standard notations

unj = u(xj , tn), unj =

1

∆x

∫ xj+1/2

xj−1/2

u(x, tn) dx.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 6/42

Space discretizations

Consider first the case of the single scalar equation

ut + f(u)x =1

εg(u).

We distinguish between schemes based on cell averages(finite volumes) and point values (finite differences).Let ∆x and ∆t be the mesh widths. We introduce the grid points

xj = j∆x, xj+1/2 = xj +1

2∆x, j = . . . ,−2,−1, 0, 1, 2, . . .

and use the standard notations

unj = u(xj , tn), unj =

1

∆x

∫ xj+1/2

xj−1/2

u(x, tn) dx.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 6/42

Finite volumes

Integrating the equation on Ij = [xj−1/2, xj+1/2] and dividingby ∆x we obtain

du

dt

∣∣∣∣j

= −1

∆x[f(u(xj+1/2, t))− f(u(xj−1/2, t)) +

1

εg(u)|j

where the bar represents the cell average.

In order to convert this expression into a numerical method we need:

To approximate the flux function at the edge of the cell, f(u(xj+1/2, t), by asuitable numerical flux function F (u(x−

j+1/2), u(x+

j+1/2)) (scalar monotone ¤ux).

To reconstruct the values u(x−j+1/2

), u(x+j+1/2

) at both sides of the right edge ofcell j, expressing the result in terms of the cell averages uj (nonoscillatorypiecewise polynomial).

To express the average of the source g(u) in terms of uj (same reconstruction).

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 7/42

Finite volumes

Integrating the equation on Ij = [xj−1/2, xj+1/2] and dividingby ∆x we obtain

du

dt

∣∣∣∣j

= −1

∆x[f(u(xj+1/2, t))− f(u(xj−1/2, t)) +

1

εg(u)|j

where the bar represents the cell average.In order to convert this expression into a numerical method we need:

To approximate the flux function at the edge of the cell, f(u(xj+1/2, t), by asuitable numerical flux function F (u(x−

j+1/2), u(x+

j+1/2)) (scalar monotone ¤ux).

To reconstruct the values u(x−j+1/2

), u(x+j+1/2

) at both sides of the right edge ofcell j, expressing the result in terms of the cell averages uj (nonoscillatorypiecewise polynomial).

To express the average of the source g(u) in terms of uj (same reconstruction).

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 7/42

Finite volumes II

Implicit time discretizations on g(u) cannot be appliedstraightforwardly since on the right hand side we have theaverage of the source term g(u) instead of the source termevaluated at the average of u, g(u).

This makes it unpractical to construct IMEX-like schemesof order higher than two. In fact

g(u) = g(u) +O(∆x2).

However, in several cases the implicit relaxation step canbe explicitly know analytically. In such cases, finite volumeschemes can be successfully employed.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 8/42

Finite volumes II

Implicit time discretizations on g(u) cannot be appliedstraightforwardly since on the right hand side we have theaverage of the source term g(u) instead of the source termevaluated at the average of u, g(u).This makes it unpractical to construct IMEX-like schemesof order higher than two. In fact

g(u) = g(u) +O(∆x2).

However, in several cases the implicit relaxation step canbe explicitly know analytically. In such cases, finite volumeschemes can be successfully employed.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 8/42

Finite differences

Basic unknown: pointwise value uj(t). It is possible to writea finite difference scheme in conservative form [S.Osher,C-W.Shu, JCP 88]

A finite difference scheme with source takes the following form:

duj

dt= − 1

h[Fj+1/2 − Fj−1/2] +

1

εg(uj)

Fj+1/2 = f+(u−j+1/2

) + f−(u+j+1/2

).

Where f+(u−j+1/2

) is obtained by:

Computing f+(ul) and interpret it as cell average of f+

Performing pointwise reconstruction of f+ in cell j, and evaluate it in xj+1/2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 9/42

Finite differences

Basic unknown: pointwise value uj(t). It is possible to writea finite difference scheme in conservative form [S.Osher,C-W.Shu, JCP 88]A finite difference scheme with source takes the following form:

duj

dt= − 1

h[Fj+1/2 − Fj−1/2] +

1

εg(uj)

Fj+1/2 = f+(u−j+1/2

) + f−(u+j+1/2

).

Where f+(u−j+1/2

) is obtained by:

Computing f+(ul) and interpret it as cell average of f+

Performing pointwise reconstruction of f+ in cell j, and evaluate it in xj+1/2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 9/42

Finite differences

Basic unknown: pointwise value uj(t). It is possible to writea finite difference scheme in conservative form [S.Osher,C-W.Shu, JCP 88]A finite difference scheme with source takes the following form:

duj

dt= − 1

h[Fj+1/2 − Fj−1/2] +

1

εg(uj)

Fj+1/2 = f+(u−j+1/2

) + f−(u+j+1/2

).

Where f+(u−j+1/2

) is obtained by:

Computing f+(ul) and interpret it as cell average of f+

Performing pointwise reconstruction of f+ in cell j, and evaluate it in xj+1/2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 9/42

Remarks

Finite volume methods can be used on arbitrary non uniform meshes. Finitedifference can be used only on uniform (or smoothly varying mesh). This makesfinite volume more flexible in several dimensions (they can even be constructed onunstructured grids).

For finite volume methods applied to systems, better results are usually obtained ifone uses characteristic variables rather than conservative variables in thereconstruction step.

There is some difference in the sharpness of the resolution of the numericalresults, according the numerical flux. Godunov flux gives much sharper results onlinear discontinuities (the difference, however, becomes less relevant with theincrease of the order of accuracy).

The general approach that we are using to treat source terms is based on themethod of lines: the space is discretized by a finite volume or finite differencemethod, and then the original system of PDE’s is approximated by a large systemof ODE’s.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 10/42

Remarks

Finite volume methods can be used on arbitrary non uniform meshes. Finitedifference can be used only on uniform (or smoothly varying mesh). This makesfinite volume more flexible in several dimensions (they can even be constructed onunstructured grids).

For finite volume methods applied to systems, better results are usually obtained ifone uses characteristic variables rather than conservative variables in thereconstruction step.

There is some difference in the sharpness of the resolution of the numericalresults, according the numerical flux. Godunov flux gives much sharper results onlinear discontinuities (the difference, however, becomes less relevant with theincrease of the order of accuracy).

The general approach that we are using to treat source terms is based on themethod of lines: the space is discretized by a finite volume or finite differencemethod, and then the original system of PDE’s is approximated by a large systemof ODE’s.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 10/42

Remarks

Finite volume methods can be used on arbitrary non uniform meshes. Finitedifference can be used only on uniform (or smoothly varying mesh). This makesfinite volume more flexible in several dimensions (they can even be constructed onunstructured grids).

For finite volume methods applied to systems, better results are usually obtained ifone uses characteristic variables rather than conservative variables in thereconstruction step.

There is some difference in the sharpness of the resolution of the numericalresults, according the numerical flux. Godunov flux gives much sharper results onlinear discontinuities (the difference, however, becomes less relevant with theincrease of the order of accuracy).

The general approach that we are using to treat source terms is based on themethod of lines: the space is discretized by a finite volume or finite differencemethod, and then the original system of PDE’s is approximated by a large systemof ODE’s.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 10/42

Remarks

Finite volume methods can be used on arbitrary non uniform meshes. Finitedifference can be used only on uniform (or smoothly varying mesh). This makesfinite volume more flexible in several dimensions (they can even be constructed onunstructured grids).

For finite volume methods applied to systems, better results are usually obtained ifone uses characteristic variables rather than conservative variables in thereconstruction step.

There is some difference in the sharpness of the resolution of the numericalresults, according the numerical flux. Godunov flux gives much sharper results onlinear discontinuities (the difference, however, becomes less relevant with theincrease of the order of accuracy).

The general approach that we are using to treat source terms is based on themethod of lines: the space is discretized by a finite volume or finite differencemethod, and then the original system of PDE’s is approximated by a large systemof ODE’s.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 10/42

Time discretizations

We consider the system of ODE’s

y′ = f(y) +1

εg(y),

where y = y(t) ∈ RN , f, g : RN → RN .

Operator Splitting:Solve separately the non-stiff advection problem and the stiff source problem

y′ = f(y), t ∈ [0, T ] y′ =1

εg(y), t ∈ [0, T ].

Although it is only first order accurate (even if the two steps are exact, unless theoperators commute), it is very popular due to its simple concept and the freedom inchoosing different solvers for advection and sources. Higher order splitting (ex. Strangsplitting) can be constructed but present a loss of accuracy when the source term is stiff.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 11/42

Time discretizations

We consider the system of ODE’s

y′ = f(y) +1

εg(y),

where y = y(t) ∈ RN , f, g : RN → RN .Operator Splitting:Solve separately the non-stiff advection problem and the stiff source problem

y′ = f(y), t ∈ [0, T ] y′ =1

εg(y), t ∈ [0, T ].

Although it is only first order accurate (even if the two steps are exact, unless theoperators commute), it is very popular due to its simple concept and the freedom inchoosing different solvers for advection and sources. Higher order splitting (ex. Strangsplitting) can be constructed but present a loss of accuracy when the source term is stiff.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 11/42

IMEX Runge-Kutta methods

An IMEX Runge-Kutta scheme has the form [U.Ascher, S.Ruth, B.Wetton ’95, U.Ascher,S.Ruth, R.Spiteri ’97]

Yi = y0 + h

i−1∑

j=1

aijf(t0 + cjh, Yj) + h

ν∑

j=1

aij1

εg(t0 + cjh, Yj),

y1 = y0 + h

ν∑

i=1

wif(t0 + cih, Yi) + h

ν∑

i=1

wi1

εg(t0 + cih, Yi).

A = (aij), aij = 0, j ≥ i and A = (aij): ν × ν matrices.

They can be represented as

Double Butcher tableau:c A

wT

c A

wT.

with c = (c1, . . . , cν)T , w = (w1, . . . , wν)T , c = (c1, . . . , cν)T , w = (w1, . . . , wν)T . Ifthe implicit scheme is diagonally implicit (DIRK), aij = 0, j > i, f is evaluated explicitly.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 12/42

IMEX Runge-Kutta methods

An IMEX Runge-Kutta scheme has the form [U.Ascher, S.Ruth, B.Wetton ’95, U.Ascher,S.Ruth, R.Spiteri ’97]

Yi = y0 + h

i−1∑

j=1

aijf(t0 + cjh, Yj) + h

ν∑

j=1

aij1

εg(t0 + cjh, Yj),

y1 = y0 + h

ν∑

i=1

wif(t0 + cih, Yi) + h

ν∑

i=1

wi1

εg(t0 + cih, Yi).

A = (aij), aij = 0, j ≥ i and A = (aij): ν × ν matrices.They can be represented as

Double Butcher tableau:c A

wT

c A

wT.

with c = (c1, . . . , cν)T , w = (w1, . . . , wν)T , c = (c1, . . . , cν)T , w = (w1, . . . , wν)T . Ifthe implicit scheme is diagonally implicit (DIRK), aij = 0, j > i, f is evaluated explicitly.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 12/42

Order conditions

Assuming ci =∑

j aij , ci =∑

j aij ,∑

i wi = 1,∑

i wi = 1,we can restrict toautonomous systems and first order requirements are satisfied.

Second order:

∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,

Third order:

∑

ij wiaij cj = 1/6,∑

i wicici = 1/3,∑

ij wiaijcj = 1/6,∑

i wicici = 1/3,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaijcj = 1/6,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaij cj = 1/6,

∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 13/42

Order conditions

Assuming ci =∑

j aij , ci =∑

j aij ,∑

i wi = 1,∑

i wi = 1,we can restrict toautonomous systems and first order requirements are satisfied.Second order:

∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,

Third order:

∑

ij wiaij cj = 1/6,∑

i wicici = 1/3,∑

ij wiaijcj = 1/6,∑

i wicici = 1/3,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaijcj = 1/6,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaij cj = 1/6,

∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 13/42

Order conditions

Assuming ci =∑

j aij , ci =∑

j aij ,∑

i wi = 1,∑

i wi = 1,we can restrict toautonomous systems and first order requirements are satisfied.Second order:

∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,∑

i wici = 1/2,

Third order:

∑

ij wiaij cj = 1/6,∑

i wicici = 1/3,∑

ij wiaijcj = 1/6,∑

i wicici = 1/3,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaijcj = 1/6,

∑

ij wiaijcj = 1/6,∑

ij wiaij cj = 1/6,∑

ij wiaij cj = 1/6,

∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3,∑

i wicici = 1/3.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 13/42

Order conditions II

If wi = wi and ci = ci, then mixed conditions are automatically satisfied. This isnot true for higher that third order accuracy

IMEX-RK schemes are a particular case of additive Runge-Kutta (ARK) methodsand therefore higher order conditions can be derived using a generalization ofButcher 1-trees to 2-trees. However the number of coupling conditions increasedramatically with the order of the schemes [M.Carpenter, C.Kennedy, ANM ’03].

IMEX-RK Number of coupling conditions

Order General case wi = wi c = c c = c and wi = wi

1 0 0 0 0

2 2 0 0 0

3 12 3 2 0

4 56 21 12 2

5 252 110 54 15

6 1128 528 218 78

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 14/42

Order conditions II

If wi = wi and ci = ci, then mixed conditions are automatically satisfied. This isnot true for higher that third order accuracy

IMEX-RK schemes are a particular case of additive Runge-Kutta (ARK) methodsand therefore higher order conditions can be derived using a generalization ofButcher 1-trees to 2-trees. However the number of coupling conditions increasedramatically with the order of the schemes [M.Carpenter, C.Kennedy, ANM ’03].

IMEX-RK Number of coupling conditions

Order General case wi = wi c = c c = c and wi = wi

1 0 0 0 0

2 2 0 0 0

3 12 3 2 0

4 56 21 12 2

5 252 110 54 15

6 1128 528 218 78

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 14/42

Relaxation systemsLet us consider an hyperbolic system with relaxation

∂tU + ∂xF (U) =1

εR(U), x ∈ R.

The operator R : RN → RN is a relaxation operator if there exists a n×N matrix Q

with rank(Q) = n < N such that QR(U) = 0 ∀ U ∈ RN .

This gives n independent conserved quantities u = QU that uniquely determine a localequilibrium U = E(u), such that R(E(u)) = 0, and that satisfy

∂t(QU) + ∂x(QF (U)) = 0.

For ε→ 0⇒ R(U) = 0⇒ U = E(u) and we have the equilibrium system

∂tu+ ∂xF(u) = 0, F(u) = QF (E(u)).

A subcharacteristic condition on F(u) has to be satisfied to ensure that u is actually thesolution of the limit equation [G.Chen, D.Levermore, T.P.Liu, CPAM ’94].

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 15/42

Relaxation systemsLet us consider an hyperbolic system with relaxation

∂tU + ∂xF (U) =1

εR(U), x ∈ R.

The operator R : RN → RN is a relaxation operator if there exists a n×N matrix Q

with rank(Q) = n < N such that QR(U) = 0 ∀ U ∈ RN .

This gives n independent conserved quantities u = QU that uniquely determine a localequilibrium U = E(u), such that R(E(u)) = 0, and that satisfy

∂t(QU) + ∂x(QF (U)) = 0.

For ε→ 0⇒ R(U) = 0⇒ U = E(u) and we have the equilibrium system

∂tu+ ∂xF(u) = 0, F(u) = QF (E(u)).

A subcharacteristic condition on F(u) has to be satisfied to ensure that u is actually thesolution of the limit equation [G.Chen, D.Levermore, T.P.Liu, CPAM ’94].

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 15/42

AP schemes

IMEX schemes for hyperbolic system with relaxations [L.P., G.Russo, J.Sci.Comp.’04]

U(i)i = U0 − h

i−1∑

j=1

aijDxF (U(j)) + h

ν∑

j=1

aij1

εR(U(j)),

U1 = U0 − h

ν∑

i=1

wiDxF (U(i)) + h

ν∑

i=1

wi1

εR(U(i)).

where DxF is a suitable discretization of ∂xF .

Definition 1: An IMEX scheme for an hyperbolic systemwith relaxation is asymptotic preserving (AP) if in the limitε→ 0 the scheme becomes a consistent discretization ofthe limit system of conservation laws. We use the notationAPk if the scheme is of order k in the limit ε→ 0.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 16/42

AP schemes

IMEX schemes for hyperbolic system with relaxations [L.P., G.Russo, J.Sci.Comp.’04]

U(i)i = U0 − h

i−1∑

j=1

aijDxF (U(j)) + h

ν∑

j=1

aij1

εR(U(j)),

U1 = U0 − h

ν∑

i=1

wiDxF (U(i)) + h

ν∑

i=1

wi1

εR(U(i)).

where DxF is a suitable discretization of ∂xF .

Definition 1: An IMEX scheme for an hyperbolic systemwith relaxation is asymptotic preserving (AP) if in the limitε→ 0 the scheme becomes a consistent discretization ofthe limit system of conservation laws. We use the notationAPk if the scheme is of order k in the limit ε→ 0.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 16/42

AP schemes II

Theorem 1 If detA 6= 0 then in the limit ε→ 0, the IMEXscheme applied to an hyperbolic system with relaxationbecomes the explicit RK scheme characterized by (A, w, c)

applied to the limit system of conservation laws.

The theorem guarantees that in the stiff limit the IMEX scheme becomes theexplicit RK scheme applied to the equilibrium system. Therefore the limiting orderof accuracy is greater or equal to the order of accuracy of the original scheme.

To satisfy detA 6= 0 the vectors of c and c cannot be equal. In fact c1 = 0 whereasc1 6= 0. If c1 = 0 then the corresponding scheme may be inaccurate if the initialcondition is not “well prepared”. This initial layer effect can be cured by usingRichardson extrapolation for the first step.

The result does not guarantee accuracy of the solution for the N − n nonconserved quantities since the last step in the scheme it is not a projection towardsthe local equilibrium. A final layer effect may occur.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 17/42

AP schemes II

Theorem 1 If detA 6= 0 then in the limit ε→ 0, the IMEXscheme applied to an hyperbolic system with relaxationbecomes the explicit RK scheme characterized by (A, w, c)

applied to the limit system of conservation laws.

The theorem guarantees that in the stiff limit the IMEX scheme becomes theexplicit RK scheme applied to the equilibrium system. Therefore the limiting orderof accuracy is greater or equal to the order of accuracy of the original scheme.

To satisfy detA 6= 0 the vectors of c and c cannot be equal. In fact c1 = 0 whereasc1 6= 0. If c1 = 0 then the corresponding scheme may be inaccurate if the initialcondition is not “well prepared”. This initial layer effect can be cured by usingRichardson extrapolation for the first step.

The result does not guarantee accuracy of the solution for the N − n nonconserved quantities since the last step in the scheme it is not a projection towardsthe local equilibrium. A final layer effect may occur.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 17/42

Stability : ε → 0

The viscosity solution of a system of conservation laws (or balance laws with arelaxation term) is decreasing in norm: ‖U(t)‖ ↘ (dissipative systems). It wouldbe desirable that the numerical solution has the same behavior.

Definition 2: A sequence {Un}n∈N is said to be stronglystable in a given norm || · || if ||Un+1|| ≤ ||Un|| for alln ≥ 0.A scheme for conservation laws is said strongly stability preserving (SSP) if itproduces a sequence which is strongly stable (for suitable ∆t < ∆tc). The mostcommonly used norms are the TV -norm (TVD property) and the infinity norm.

As a consequence if the explicit part of the IMEX scheme is SSP then, as ε→ 0,one obtains a SSP method for the limiting conservation law. This asymptotic SSPproperty is essential to avoid spurious oscillations in the limit scheme for thelimiting system of conservation laws.

[S.Gottlieb, C-W.Shu, E.Tadmor SIREV ’01, R.Spiteri, S.Ruth, SINUM ’02 ]

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 18/42

Stability : ε → 0

The viscosity solution of a system of conservation laws (or balance laws with arelaxation term) is decreasing in norm: ‖U(t)‖ ↘ (dissipative systems). It wouldbe desirable that the numerical solution has the same behavior.

Definition 2: A sequence {Un}n∈N is said to be stronglystable in a given norm || · || if ||Un+1|| ≤ ||Un|| for alln ≥ 0.

A scheme for conservation laws is said strongly stability preserving (SSP) if itproduces a sequence which is strongly stable (for suitable ∆t < ∆tc). The mostcommonly used norms are the TV -norm (TVD property) and the infinity norm.

As a consequence if the explicit part of the IMEX scheme is SSP then, as ε→ 0,one obtains a SSP method for the limiting conservation law. This asymptotic SSPproperty is essential to avoid spurious oscillations in the limit scheme for thelimiting system of conservation laws.

[S.Gottlieb, C-W.Shu, E.Tadmor SIREV ’01, R.Spiteri, S.Ruth, SINUM ’02 ]

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 18/42

Stability : ε → 0

The viscosity solution of a system of conservation laws (or balance laws with arelaxation term) is decreasing in norm: ‖U(t)‖ ↘ (dissipative systems). It wouldbe desirable that the numerical solution has the same behavior.

Definition 2: A sequence {Un}n∈N is said to be stronglystable in a given norm || · || if ||Un+1|| ≤ ||Un|| for alln ≥ 0.A scheme for conservation laws is said strongly stability preserving (SSP) if itproduces a sequence which is strongly stable (for suitable ∆t < ∆tc). The mostcommonly used norms are the TV -norm (TVD property) and the infinity norm.

As a consequence if the explicit part of the IMEX scheme is SSP then, as ε→ 0,one obtains a SSP method for the limiting conservation law. This asymptotic SSPproperty is essential to avoid spurious oscillations in the limit scheme for thelimiting system of conservation laws.

[S.Gottlieb, C-W.Shu, E.Tadmor SIREV ’01, R.Spiteri, S.Ruth, SINUM ’02 ]

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 18/42

Stability : ε → 0

The viscosity solution of a system of conservation laws (or balance laws with arelaxation term) is decreasing in norm: ‖U(t)‖ ↘ (dissipative systems). It wouldbe desirable that the numerical solution has the same behavior.

Definition 2: A sequence {Un}n∈N is said to be stronglystable in a given norm || · || if ||Un+1|| ≤ ||Un|| for alln ≥ 0.A scheme for conservation laws is said strongly stability preserving (SSP) if itproduces a sequence which is strongly stable (for suitable ∆t < ∆tc). The mostcommonly used norms are the TV -norm (TVD property) and the infinity norm.

As a consequence if the explicit part of the IMEX scheme is SSP then, as ε→ 0,one obtains a SSP method for the limiting conservation law. This asymptotic SSPproperty is essential to avoid spurious oscillations in the limit scheme for thelimiting system of conservation laws.

[S.Gottlieb, C-W.Shu, E.Tadmor SIREV ’01, R.Spiteri, S.Ruth, SINUM ’02 ]

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 18/42

Stability analysisTo study the A-stability of a RK scheme, one considers a scalar equation of the form

y′ = λy, y(0) = 1

with y : R+ → R, and λ ∈ R,<λ ≤ 0. Such test problem is sufficient to characterize thestability property of a Runge-Kutta scheme when applied to a linear system of the form

y′ = By, y(0) = y0

with y ∈ Rm, and B ∈ Rm×m.

For IMEX-RK, and in general for additive Runge-Kutta methods, such a stability analysishas a limited validity. In fact, consider a generic linear system of the form

y′ = B1 y +B2 y (1)

with y ∈ Rm, and B1, B2 ∈ Rm×m, and apply an IMEX scheme which is explicit in B1y

and implicit in B2y. The stability of the numerical solution depends on the two matricesB1 and B2, and not only on their eigenvalues since in general the two matrices do notshare the same eigenvectors, and therefore they can not be diagonalized simultaneously.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 19/42

Stability analysisTo study the A-stability of a RK scheme, one considers a scalar equation of the form

y′ = λy, y(0) = 1

with y : R+ → R, and λ ∈ R,<λ ≤ 0. Such test problem is sufficient to characterize thestability property of a Runge-Kutta scheme when applied to a linear system of the form

y′ = By, y(0) = y0

with y ∈ Rm, and B ∈ Rm×m.

For IMEX-RK, and in general for additive Runge-Kutta methods, such a stability analysishas a limited validity. In fact, consider a generic linear system of the form

y′ = B1 y +B2 y (2)

with y ∈ Rm, and B1, B2 ∈ Rm×m, and apply an IMEX scheme which is explicit in B1y

and implicit in B2y. The stability of the numerical solution depends on the two matricesB1 and B2, and not only on their eigenvalues since in general the two matrices do notshare the same eigenvectors, and therefore they can not be diagonalized simultaneously.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 19/42

Stability matrix for systems

Let us consider an IMEX-RK (or a partitioned RK) scheme applied to system (1).After some manipulation the scheme can be conveniently written as

yn+1 = R(Z1, Z2)yn

where Z1 = hB1, Z2 = hB2, and the matrix of absolute stability is given by

R(Z1, Z2) = Im + (wT ⊗ Z1 + wT ⊗ Z2)(Iνm − A⊗ Z1 −A⊗ Z2)−1e⊗ Im

with e ≡ (1, . . . , 1)T ∈ Rm and the Kronecker product is defined as

e⊗ yn =

yn

...

yn

, A⊗B =

a11B · · · a1νB

.... . .

...

aν1B · · · aννB

The spectral radius of R(Z1, Z2) does not depend only on the eigenvalues of Z1, Z2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 20/42

Example

Consider simple 2× 2 relaxation system is given by

ut + vx = 0

vt + ux = −µ(v − bu)

In the stiff limit µ→∞ for 0 < b < 1 the system relaxes to ut + bux = 0

Looking for a Fourier solution u = u(t)eiξx, v = v(t)eiξx, the system can be written as

dU

dt= C1U + C2U

U =

u

v

, C1 = −iξ

0 1

1 0

, C2 = −µ

0 0

−b 1

The boundary of the region of absolute stability is given by the relationρ(R(Z1, Z2)) = 1 where ρ denotes the spectral radius, and Z1 = hC1, Z2 = hC2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 21/42

Example

Consider simple 2× 2 relaxation system is given by

ut + vx = 0

vt + ux = −µ(v − bu)

In the stiff limit µ→∞ for 0 < b < 1 the system relaxes to ut + bux = 0

Looking for a Fourier solution u = u(t)eiξx, v = v(t)eiξx, the system can be written as

dU

dt= C1U + C2U

U =

u

v

, C1 = −iξ

0 1

1 0

, C2 = −µ

0 0

−b 1

The boundary of the region of absolute stability is given by the relationρ(R(Z1, Z2)) = 1 where ρ denotes the spectral radius, and Z1 = hC1, Z2 = hC2.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 21/42

Stability regions: first order

10−2 10−1 100 101 102 103 104 105 106−5

−4

−3

−2

−1

0

1

2

3

4

5

µ h

ξ h

SP−111

b = 0.1b = 0.3b = 0.5b = 0.7b = 0.9

Implicit-Explicit Euler in the ξh–µh plane, for several values of the parameter b

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 22/42

Stability regions: second order

10−2 10−1 100 101 102 103 104 105 106−5

−4

−3

−2

−1

0

1

2

3

4

5

µ h

ξ h

SSP2−222

b = 0.1b = 0.3b = 0.5b = 0.7b = 0.9

Second order IMEX-SSP scheme in the ξh–µh plane.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 23/42

Stability regions: third order

10−2 10−1 100 101 102 103 104 105 106−5

−4

−3

−2

−1

0

1

2

3

4

5

µ h

ξ h

SSP3−332

b = 0.1b = 0.3b = 0.5b = 0.7b = 0.9

Third order IMEX-SSP scheme in the ξh–µh plane.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 24/42

A numerical example

We consider the Broadwell model equations

∂tρ + ∂xm = 0,

∂tm + ∂xz = 0,

∂tz + ∂xm =1

ε(ρ2 + m2 − 2ρz),

where ε is the mean free path. The dynamical variables ρ

and m are the density and the momentum respectively,while z represents the flux of momentum.

We perform an accuracy test for second order IMEX-SSPschemes with smooth initial data and periodic b.c.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 25/42

A numerical example

We consider the Broadwell model equations

∂tρ + ∂xm = 0,

∂tm + ∂xz = 0,

∂tz + ∂xm =1

ε(ρ2 + m2 − 2ρz),

where ε is the mean free path. The dynamical variables ρ

and m are the density and the momentum respectively,while z represents the flux of momentum.We perform an accuracy test for second order IMEX-SSPschemes with smooth initial data and periodic b.c.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 25/42

Convergence Rates

ε 1.0 10−1 10−2 10−3 10−5

Convergence Rates for ρ N

2.07772 2.11211 1.96649 2.04715 2.06860 100-200

2.04453 2.07418 2.00709 1.98219 2.04030 200-400

Convergence Rates for z

2.06926 2.23039 1.60070 1.85409 2.08532 100-200

2.03151 2.17488 1.76238 1.59695 2.04047 200-400

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 26/42

Applications: Traffic flow

We consider the macroscopic model [A.Aw, M.Rascle SIAPP ’00]

∂tρ+ ∂x(ρv) = 0,

∂t(ρw) + ∂x(vρw) =ρ

T(V (ρ)− v),

where w = v + P (ρ), P (ρ) describes the anticipation of road conditions in front of thedrivers, V (ρ) describes the Fundamental Diagram and T is the relaxation time.

If the relaxation time goes to zero, under the subcharacteristic condition−P ′(ρ) ≤ V ′(ρ) ≤ 0, ρ > 0, we obtain the Lighthill-Whitham model

∂tρ+ ∂x(ρV (ρ)) = 0.

We take P (ρ) = cv ln(

ρρm

)

, where ρm is a given maximal density and cv a constant.

We assume V (ρ) fitting to some experimental data [Aw, Klar, Materne, Rascle]. Weassume cv = 2. All quantities are normalized so that vm = 1 and ρm = 1.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 27/42

Applications: Traffic flow

We consider the macroscopic model [A.Aw, M.Rascle SIAPP ’00]

∂tρ+ ∂x(ρv) = 0,

∂t(ρw) + ∂x(vρw) =ρ

T(V (ρ)− v),

where w = v + P (ρ), P (ρ) describes the anticipation of road conditions in front of thedrivers, V (ρ) describes the Fundamental Diagram and T is the relaxation time.If the relaxation time goes to zero, under the subcharacteristic condition−P ′(ρ) ≤ V ′(ρ) ≤ 0, ρ > 0, we obtain the Lighthill-Whitham model

∂tρ+ ∂x(ρV (ρ)) = 0.

We take P (ρ) = cv ln(

ρρm

)

, where ρm is a given maximal density and cv a constant.

We assume V (ρ) fitting to some experimental data [Aw, Klar, Materne, Rascle]. Weassume cv = 2. All quantities are normalized so that vm = 1 and ρm = 1.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 27/42

Applications: Traffic flow

We consider the macroscopic model [A.Aw, M.Rascle SIAPP ’00]

∂tρ+ ∂x(ρv) = 0,

∂t(ρw) + ∂x(vρw) =ρ

T(V (ρ)− v),

where w = v + P (ρ), P (ρ) describes the anticipation of road conditions in front of thedrivers, V (ρ) describes the Fundamental Diagram and T is the relaxation time.If the relaxation time goes to zero, under the subcharacteristic condition−P ′(ρ) ≤ V ′(ρ) ≤ 0, ρ > 0, we obtain the Lighthill-Whitham model

∂tρ+ ∂x(ρV (ρ)) = 0.

We take P (ρ) = cv ln(

ρρm

)

, where ρm is a given maximal density and cv a constant.

We assume V (ρ) fitting to some experimental data [Aw, Klar, Materne, Rascle]. Weassume cv = 2. All quantities are normalized so that vm = 1 and ρm = 1.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 27/42

Traffic flow II

We consider a Riemann problem centered at x = 0

ρL = 0.05, vL = 0.05, ρR = 0.05, vR = 0.5.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.50.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

x

ρ(x,

t), ρ

v(x

,t)

IMEX−SSP2−WENO, ε=0.2, t=1, N=200

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.50.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

x

ρ(x,

t), ρ

v(x

,t)

IMEX−SSP3−WENO, ε=0.2, t=1, N=200

Solution at t = 1 for T = 0.2 for ρ(◦) and ρv(∗) at time t = 1 for ε = 0.2. Left secondorder scheme, right third order scheme

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 28/42

Granular gases

We consider a hydrodynamic model of a granular gas [J.Jenkins, M.Richman, ARMA ’87]

ρt + (ρu)x = 0,

(ρu)t + (ρu2 + p)x = ρg,(1

2ρu2 +

3

2ρT

)

t

+

(1

2ρu3 +

3

2uρT + pu

)

x

= − (1− e2)

εG(ρ)ρ2T 3/2,

where e is the coefficient of restitution, g the acceleration due to gravity, ε a relaxationtime, p is the pressure given by p = ρT (1 + 2(1 + e)G(ρ)), and G(ρ) is the statisticalcorrelation function.

In our experiments we assume

G(ρ) = ν

(

1−(

ν

νM

) 43νM)−1

,

ν = σ3ρπ/6: volume fraction, σ: diameter of a particle, νM = 0.64994: is 3Dclose-packed constant.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 29/42

Granular gases

We consider a hydrodynamic model of a granular gas [J.Jenkins, M.Richman, ARMA ’87]

ρt + (ρu)x = 0,

(ρu)t + (ρu2 + p)x = ρg,(1

2ρu2 +

3

2ρT

)

t

+

(1

2ρu3 +

3

2uρT + pu

)

x

= − (1− e2)

εG(ρ)ρ2T 3/2,

where e is the coefficient of restitution, g the acceleration due to gravity, ε a relaxationtime, p is the pressure given by p = ρT (1 + 2(1 + e)G(ρ)), and G(ρ) is the statisticalcorrelation function.In our experiments we assume

G(ρ) = ν

(

1−(

ν

νM

) 43νM)−1

,

ν = σ3ρπ/6: volume fraction, σ: diameter of a particle, νM = 0.64994: is 3Dclose-packed constant.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 29/42

Granular gases II

We consider the following initial data [A.Marquina, S.Serna’04] on the interval [0, 10]

ρ = 34.37746770, v = 18, P = 1589.2685472

which corresponds to a supersonic flow at Mach number Ma = 7 (the ratio of the meanfluid speed to the speed of sound). Zero-flux boundary condition on the bottom, ingoingflow on the top. Restitution coefficient e = 0.97, particle diameter σ = 0.1.

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

ν(x,

t)

IMEX−SSP2−WENO, ε=0.01, t=0.2, N=200

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

ν(x,

t)

IMEX−SSP3−WENO, ε=0.01, t=0.2, N=200

Solution for ν at t = 0.2 with ε = 0.01, second (left) and third (right) order schemes

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 30/42

Remarks

Lack of analysis of the uniform accuracy in ε for the general case.Further conditions should be imposed (severe restrictions).Conditions such that IMEX schemes are accurate for the O(ε)

term (capture the diffusion limit) have been derived recently [L.P.,G.Russo ’04].

IMEX schemes with SSP property have been derived up to thirdorder. Possible extension to higher order.

Taking advantage of the stabilizing effect of the relaxation terms toreduce the amount of numerical dissipation in the discretization ofthe convection part for non stiff values of ε.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 31/42

Remarks

Lack of analysis of the uniform accuracy in ε for the general case.Further conditions should be imposed (severe restrictions).Conditions such that IMEX schemes are accurate for the O(ε)

term (capture the diffusion limit) have been derived recently [L.P.,G.Russo ’04].

IMEX schemes with SSP property have been derived up to thirdorder. Possible extension to higher order.

Taking advantage of the stabilizing effect of the relaxation terms toreduce the amount of numerical dissipation in the discretization ofthe convection part for non stiff values of ε.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 31/42

Remarks

Lack of analysis of the uniform accuracy in ε for the general case.Further conditions should be imposed (severe restrictions).Conditions such that IMEX schemes are accurate for the O(ε)

term (capture the diffusion limit) have been derived recently [L.P.,G.Russo ’04].

IMEX schemes with SSP property have been derived up to thirdorder. Possible extension to higher order.

Taking advantage of the stabilizing effect of the relaxation terms toreduce the amount of numerical dissipation in the discretization ofthe convection part for non stiff values of ε.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 31/42

Hybrid methodsIn many applications the computational cost of solving the system ofequations can be very high (kinetic modelling, mathematical finance,biomathematics, computer graphics, . . .). A probabilistic approach ishighly desirable.

Monte Carlo methods are widely used to simulate complex systems. Theyhave many advantages in terms of computational cost, physical properties(thanks to the particle interpretation of the sample), simplicity whendealing with complicate geometries.

In stiff relaxation-like systems we often have a strong dimension reductionof the problem. For the reduced hyperbolic systems it would be natural tokeep a deterministic (finite volume/differences) approximation.

The design of such hybrid methodology often involveshybrid-modelling/multi-modelling and its details are rather problemdependent.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 32/42

Hybrid methodsIn many applications the computational cost of solving the system ofequations can be very high (kinetic modelling, mathematical finance,biomathematics, computer graphics, . . .). A probabilistic approach ishighly desirable.

Monte Carlo methods are widely used to simulate complex systems. Theyhave many advantages in terms of computational cost, physical properties(thanks to the particle interpretation of the sample), simplicity whendealing with complicate geometries.

In stiff relaxation-like systems we often have a strong dimension reductionof the problem. For the reduced hyperbolic systems it would be natural tokeep a deterministic (finite volume/differences) approximation.

The design of such hybrid methodology often involveshybrid-modelling/multi-modelling and its details are rather problemdependent.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 32/42

Hybrid methodsIn many applications the computational cost of solving the system ofequations can be very high (kinetic modelling, mathematical finance,biomathematics, computer graphics, . . .). A probabilistic approach ishighly desirable.

Monte Carlo methods are widely used to simulate complex systems. Theyhave many advantages in terms of computational cost, physical properties(thanks to the particle interpretation of the sample), simplicity whendealing with complicate geometries.

In stiff relaxation-like systems we often have a strong dimension reductionof the problem. For the reduced hyperbolic systems it would be natural tokeep a deterministic (finite volume/differences) approximation.

The design of such hybrid methodology often involveshybrid-modelling/multi-modelling and its details are rather problemdependent.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 32/42

Hybrid methodsIn many applications the computational cost of solving the system ofequations can be very high (kinetic modelling, mathematical finance,biomathematics, computer graphics, . . .). A probabilistic approach ishighly desirable.

Monte Carlo methods are widely used to simulate complex systems. Theyhave many advantages in terms of computational cost, physical properties(thanks to the particle interpretation of the sample), simplicity whendealing with complicate geometries.

In stiff relaxation-like systems we often have a strong dimension reductionof the problem. For the reduced hyperbolic systems it would be natural tokeep a deterministic (finite volume/differences) approximation.

The design of such hybrid methodology often involveshybrid-modelling/multi-modelling and its details are rather problemdependent.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 32/42

The Hybrid Monte Carlo methodWe restrict here to the case of hyperbolic relaxation systems. We recall thatU(x, t) ∈ RN denotes the solution of the system whereas E(u(x, t)) ∈ RN denotes theequilibrium solution where u(x, t) ∈ Rn are the conserved variables.

We make the following ansatz

U(x, t) = (1− β(t))Up(x, t)︸ ︷︷ ︸

nonequilibrium

+ β(t)E(u(x, t))︸ ︷︷ ︸

equilibrium

,

where 0 ≤ β(t) ≤ 1 characterizes the equilibrium fractionand Up(x, t) the non equilibrium part of the solution.

Solve the evolution of the non equilibrium part by Monte Carlo methods. ThusUp(x, t) will be represented by a set of particles in the computational domain.

Solve the evolution of the equilibrium part by deterministic methods. ThusE(u(x, t)) will be represented on a suitable grid in the computational domain.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 33/42

The Hybrid Monte Carlo methodWe restrict here to the case of hyperbolic relaxation systems. We recall thatU(x, t) ∈ RN denotes the solution of the system whereas E(u(x, t)) ∈ RN denotes theequilibrium solution where u(x, t) ∈ Rn are the conserved variables.

We make the following ansatz

U(x, t) = (1− β(t))Up(x, t)︸ ︷︷ ︸

nonequilibrium

+ β(t)E(u(x, t))︸ ︷︷ ︸

equilibrium

,

where 0 ≤ β(t) ≤ 1 characterizes the equilibrium fractionand Up(x, t) the non equilibrium part of the solution.

Solve the evolution of the non equilibrium part by Monte Carlo methods. ThusUp(x, t) will be represented by a set of particles in the computational domain.

Solve the evolution of the equilibrium part by deterministic methods. ThusE(u(x, t)) will be represented on a suitable grid in the computational domain.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 33/42

The Hybrid Monte Carlo methodWe restrict here to the case of hyperbolic relaxation systems. We recall thatU(x, t) ∈ RN denotes the solution of the system whereas E(u(x, t)) ∈ RN denotes theequilibrium solution where u(x, t) ∈ Rn are the conserved variables.

We make the following ansatz

U(x, t) = (1− β(t))Up(x, t)︸ ︷︷ ︸

nonequilibrium

+ β(t)E(u(x, t))︸ ︷︷ ︸

equilibrium

,

where 0 ≤ β(t) ≤ 1 characterizes the equilibrium fractionand Up(x, t) the non equilibrium part of the solution.

Solve the evolution of the non equilibrium part by Monte Carlo methods. ThusUp(x, t) will be represented by a set of particles in the computational domain.

Solve the evolution of the equilibrium part by deterministic methods. ThusE(u(x, t)) will be represented on a suitable grid in the computational domain.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 33/42

A simple relaxation systemWe describe the method in the case of Jin-Xin relaxation system

∂tu+ ∂xv = 0

∂tv + a∂xu = −1

ε(v − F (u)).

As ε→ 0, under the subcharacteristic condition a > f ′(u)2 we obtain the scalarconservation law ut + ∂xF (u) = 0.

We rewrite the system in diagonal form

∂tf +√a∂xf = −1

ε(f − Ef (u))

∂tg −√a∂xg = −1

ε(g − Eg(u)).

f =

√au+ v

2√a

, g =

√au− v

2√a

, Ef (u) =

√au+ F (u)

2√a

, Eg(u) =

√au− F (u)

2√a

.

For simplicity we assume −u√a ≤ F (U) ≤ u√a and u ≥ 0 so that f, g ≥ 0.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 34/42

A simple relaxation systemWe describe the method in the case of Jin-Xin relaxation system

∂tu+ ∂xv = 0

∂tv + a∂xu = −1

ε(v − F (u)).

As ε→ 0, under the subcharacteristic condition a > f ′(u)2 we obtain the scalarconservation law ut + ∂xF (u) = 0.

We rewrite the system in diagonal form

∂tf +√a∂xf = −1

ε(f − Ef (u))

∂tg −√a∂xg = −1

ε(g − Eg(u)).

f =

√au+ v

2√a

, g =

√au− v

2√a

, Ef (u) =

√au+ F (u)

2√a

, Eg(u) =

√au− F (u)

2√a

.

For simplicity we assume −u√a ≤ F (U) ≤ u√a and u ≥ 0 so that f, g ≥ 0.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 34/42

HMC method

We start by splitting the system in the two separate steps

Relaxation

∂tf = − 1ε(f − Ef (u))

∂tg = − 1ε(g − Eg(u))

Convection

∂tf +√a∂xf = 0

∂tg −√a∂xg = 0

Assuming

f(x, t) = (1− β(t))fp(x, t) + β(t)Ef (u), g(x, t) = (1− β(t))gp(x, t) + β(t)Eg(u)

from the exact solution of the relaxation step

f(x, t) = e−t/εf(x, 0)+(1−e−t/ε)Ef (u), g(x, t) = e−t/εg(x, 0)+(1−e−t/ε)Eg(u),

we obtain evolution equations for fp(x, t) and β(t)

fp(x, t) = e−t/ε1− β(t)

1− β(0)fp(x, 0), β(t) = e−t/εβ(0) + 1− e−t/ε.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 35/42

HMC method

We start by splitting the system in the two separate steps

Relaxation

∂tf = − 1ε(f − Ef (u))

∂tg = − 1ε(g − Eg(u))

Convection

∂tf +√a∂xf = 0

∂tg −√a∂xg = 0

Assuming

f(x, t) = (1− β(t))fp(x, t) + β(t)Ef (u), g(x, t) = (1− β(t))gp(x, t) + β(t)Eg(u)

from the exact solution of the relaxation step

f(x, t) = e−t/εf(x, 0)+(1−e−t/ε)Ef (u), g(x, t) = e−t/εg(x, 0)+(1−e−t/ε)Eg(u),

we obtain evolution equations for fp(x, t) and β(t)

fp(x, t) = e−t/ε1− β(t)

1− β(0)fp(x, 0), β(t) = e−t/εβ(0) + 1− e−t/ε.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 35/42

HMC method II

Note that β(t)→ 1 as ε→ 0. If we start from β(0) = 0 (all particles) at the end of therelaxation a fraction 1− e−t/ε of the particles is discarded by the method as the effect ofthe relaxation to equilibrium.

After relaxation the transport step takes the form

∂t(1− β)fp +√a∂x(1− β)fp + ∂tβEf (u) +

√a∂xβEf (u) = 0

∂t(1− β)gp −√a∂x(1− β)gp + ∂tβEg(u)−

√a∂xβEg(u) = 0

The convection part for fp can be solved exactly by transport of particles whereasthe convection part for the equilibrium fraction is solved by finite differences(volumes). Unfortunately the transport destroys the particle-equilibrium structureof the solution.

The simplest way to project the final numerical solution to the form required by therelaxation step is to set β = 0 and to resample the whole deterministic fraction. Inpractice since a fraction 1− e−t/ε will be discarded by the relaxation we resampleonly e−t/ε particles, thus avoiding inefficient discard-resample procedures.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 36/42

HMC method II

Note that β(t)→ 1 as ε→ 0. If we start from β(0) = 0 (all particles) at the end of therelaxation a fraction 1− e−t/ε of the particles is discarded by the method as the effect ofthe relaxation to equilibrium.After relaxation the transport step takes the form

∂t(1− β)fp +√a∂x(1− β)fp + ∂tβEf (u) +

√a∂xβEf (u) = 0

∂t(1− β)gp −√a∂x(1− β)gp + ∂tβEg(u)−

√a∂xβEg(u) = 0

The convection part for fp can be solved exactly by transport of particles whereasthe convection part for the equilibrium fraction is solved by finite differences(volumes). Unfortunately the transport destroys the particle-equilibrium structureof the solution.

The simplest way to project the final numerical solution to the form required by therelaxation step is to set β = 0 and to resample the whole deterministic fraction. Inpractice since a fraction 1− e−t/ε will be discarded by the relaxation we resampleonly e−t/ε particles, thus avoiding inefficient discard-resample procedures.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 36/42

HMC method II

Note that β(t)→ 1 as ε→ 0. If we start from β(0) = 0 (all particles) at the end of therelaxation a fraction 1− e−t/ε of the particles is discarded by the method as the effect ofthe relaxation to equilibrium.After relaxation the transport step takes the form

∂t(1− β)fp +√a∂x(1− β)fp + ∂tβEf (u) +

√a∂xβEf (u) = 0

∂t(1− β)gp −√a∂x(1− β)gp + ∂tβEg(u)−

√a∂xβEg(u) = 0

The convection part for fp can be solved exactly by transport of particles whereasthe convection part for the equilibrium fraction is solved by finite differences(volumes). Unfortunately the transport destroys the particle-equilibrium structureof the solution.

The simplest way to project the final numerical solution to the form required by therelaxation step is to set β = 0 and to resample the whole deterministic fraction. Inpractice since a fraction 1− e−t/ε will be discarded by the relaxation we resampleonly e−t/ε particles, thus avoiding inefficient discard-resample procedures.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 36/42

Numerical example: non stiff regimes

Initial data Gaussian profile and β = 0. Here F (u) = u2/2 (ε→ 0 Burgers equation).

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Mass=0.997524 Time=2.010050 N=9498 eta=0.047121

x

u(x,

t)

ExactHybridParticleContinuum

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Mass=0.996730 Time=2.010050 N=6631 eta=0.343905

x

u(x,

t)

ExactHybridParticleContinuum

Solution at t = 2 for ε = 1 (left) and ε = 0.1 (right).

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 37/42

Numerical example: stiff regimes

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Mass=0.998175 Time=2.010050 N=1665 eta=0.825820

x

u(x,

t)

ExactHybridParticleContinuum

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Mass=0.999591 Time=2.010050 N=197 eta=0.982277

xu(

x,t)

ExactHybridParticleContinuum

Solution at t = 2 for ε = 0.01 (left) and ε = 0.001 (right).

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 38/42

Numerical example: shock formation

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Mass=0.999767 Time=10.000000 N=17 eta=0.998236

x

u(x,

t)

ExactHybridParticleContinuum

Solution at t = 10 for ε = 0.001.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 39/42

Kinetic equationsThe HMC method just described can be extended directly to kinetic equations of the form

∂tf + v∇xf =1

εQ(f, f).

Here f = f(x, v, t) ≥ 0, x, v ∈ Rd and Q(f, f) describes interactions between particles.For BGK-like collision terms there are no additional difficulties.

For more general operators, after a splitting, the difficulty relies in the approximation ofthe relaxation step

∂tf =1

εQ(f, f).

For small ε we have a stiff source which originates a dense nonlinear system if solvedby implicit schemes. For Fokker-Planck-Landau like operators the implicit system can besolved efficiently by Krylov-like iterative solvers. For Boltzmann like equations howeverthe computational cost is prohibitive. To this goal we adopted the Time Relaxed (TR)schemes based on the Wild sum expansion.

[ L.P.,R.E.Caflisch JCP ’99; L.P., G.Russo SISC ’01; L.P., S.Trazzi IJNMF to appear,R.E.Caflisch, H.Luo ’04].

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 40/42

Kinetic equationsThe HMC method just described can be extended directly to kinetic equations of the form

∂tf + v∇xf =1

εQ(f, f).

Here f = f(x, v, t) ≥ 0, x, v ∈ Rd and Q(f, f) describes interactions between particles.For BGK-like collision terms there are no additional difficulties.For more general operators, after a splitting, the difficulty relies in the approximation ofthe relaxation step

∂tf =1

εQ(f, f).

For small ε we have a stiff source which originates a dense nonlinear system if solvedby implicit schemes. For Fokker-Planck-Landau like operators the implicit system can besolved efficiently by Krylov-like iterative solvers. For Boltzmann like equations howeverthe computational cost is prohibitive. To this goal we adopted the Time Relaxed (TR)schemes based on the Wild sum expansion.[ L.P.,R.E.Caflisch JCP ’99; L.P., G.Russo SISC ’01; L.P., S.Trazzi IJNMF to appear,R.E.Caflisch, H.Luo ’04].

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 40/42

Remarks

Nonlinear flux terms can be treated similarly using a relaxationapproximations by particles [L.P, M.Seaid Lect. Not. Com. Sci. ’04].

Extension to systems in the general case is difficult (lack of a generaldiscrete kinetic approximation for general systems).

The methods presented were limited to first order accuracy in space andtime. Higher order accuracy would be desirable for small ε. This can beachieved with a modification of the method taking u as a high orderapproximation of the corresponding equilibrium system.

High order IMEX methods are difficult to use in this formulation due to thepresence of the stochastic fraction.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 42/42

Remarks

Nonlinear flux terms can be treated similarly using a relaxationapproximations by particles [L.P, M.Seaid Lect. Not. Com. Sci. ’04].

Extension to systems in the general case is difficult (lack of a generaldiscrete kinetic approximation for general systems).

The methods presented were limited to first order accuracy in space andtime. Higher order accuracy would be desirable for small ε. This can beachieved with a modification of the method taking u as a high orderapproximation of the corresponding equilibrium system.

High order IMEX methods are difficult to use in this formulation due to thepresence of the stochastic fraction.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 42/42

Remarks

Nonlinear flux terms can be treated similarly using a relaxationapproximations by particles [L.P, M.Seaid Lect. Not. Com. Sci. ’04].

Extension to systems in the general case is difficult (lack of a generaldiscrete kinetic approximation for general systems).

The methods presented were limited to first order accuracy in space andtime. Higher order accuracy would be desirable for small ε. This can beachieved with a modification of the method taking u as a high orderapproximation of the corresponding equilibrium system.

High order IMEX methods are difficult to use in this formulation due to thepresence of the stochastic fraction.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 42/42

Remarks

Nonlinear flux terms can be treated similarly using a relaxationapproximations by particles [L.P, M.Seaid Lect. Not. Com. Sci. ’04].

Extension to systems in the general case is difficult (lack of a generaldiscrete kinetic approximation for general systems).

The methods presented were limited to first order accuracy in space andtime. Higher order accuracy would be desirable for small ε. This can beachieved with a modification of the method taking u as a high orderapproximation of the corresponding equilibrium system.

High order IMEX methods are difficult to use in this formulation due to thepresence of the stochastic fraction.

Lorenzo Pareschi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p. 42/42