Lecture 2

Finite Difference (FD) Methods

Conservation Law:

1D Linear Differential Equation:

0)J(U

(t,x)

t(t,x)

0UU

x(t,x)a

t(t,x)

(t,x)a(t,x) UJ

FD Methods

One-sided backward Leapfrog

One-sided forward Lax-Wendroff

Lax-Friedrichs Beam-Warming

One-sided backward

Difference equation:

)(i,j(i,j)-AΔxΔt(i,j) - ,j)(i 1UUU1U

i

i+1

i-1

j-1 j j+1

time

stepping

One-sided forward

Difference equation:

(i,j))-(i,jAΔxΔt(i,j) - ,j)(i U1UU1U

i

i+1

i-1

j-1 j j+1

time

stepping

Lax-Friedrichs

Difference equation:

)(i,j)-(i,jAΔxΔt

- )(i,j)(i,j,j)(i

1U1U2

1U1U211U

Leapfrog

Difference equation:

)(i,j)-(i,jAΔxΔt

,j)(i,j)(i

1U1U2

1U1U

Lax-Wendroff

Difference equation:

)(i,j(i,j))-(i,jAΔxΔt

)(i,j)-(i,jAΔxΔt

(i,j),j)(i

1UU21U2

1U1U2

U1U

22

2

Beam-Warming

Difference equation:

)(i,j)(i,j(i,j)-AΔxΔt

)(i,j)(i,j(i,j)-AΔxΔt

(i,j),j)(i

2U1U2U2

2U1U43U2

U1U

22

2

Time stepping

1st order methods in time:U(i+1) = F{ U(i) }

2nd order methods in time:U(i+1) = F{ U(i) , U(i-1) }

Starting A) U(2) = F{ U(1) } ( e.g. One-Sided )

method: B) U(3) = F{ U(2) , U(1) } ( Leapfrog )

Convergence

- Analytical solution- Numerical solution

Error function:

Numerical convergence (Norm of the Error function):

(t,x)u(i,j)U

))(),(()UE jxitu(i,j(i,j)

0,0E x (i,j)

Norms

Norm for conservation laws:

Other Norms (e.g. spectral problems)

dxu(x,y) u(x,y) ||

N

j

E(i,j) E(i,j)1

1

2

12

N

j

E(i,j) E(i,j)

Numerical Stability

Courant, Friedrichs, Levy (CFL, Courant number)

Upwind schemes for discontinity:

a > 0 : One sided forward a < 0 : One sided backward

txaCFL max

Lax Equivalence TheoremFor a well-posed linear initial value problem, the

method is convergent if and only if it is stable.

Necessary and sufficient condition for consistent linear method

Well posed: solution exists, continuous, unique;

- numerical stability (t)- numerical convergence (xyz)

Discontinuous solutionsAdvection equation:

Initial condition:

Analytic solution:

0UU

x(t,x)a

t(t,x)

0001

)(U0 xx

x

atxxt 0U),(U

Analytic vs Numerical

x = 0.01

Analytic vs Numerical

x = 0.001

Nonlinear EquationsLinear conservation:

Nonlinear conservation:

0UU

x(t,x)a

t(t,x)

0)U(U

Fxt

(t,x)

)U(U Fxx

(t,x)a

Nonlinear FD methodsLax-Friedrichslinear stencil:

nonlinear stencil:

)(i,j)-(i,jaΔxΔt

)(i,j)(i,j,j)(i

1U1U2

1U1U211U

))(i,j))-F((i,jF(ΔxΔt

)(i,j)(i,j,j)(i

1U1U2

1U1U211U

Nonlinear FD methods

Lax-Wendroff

MacCormack’s two step method:

)(i,j(i,j))-(i,jAΔxΔt

)(i,j)-(i,jAΔxΔt

(i,j),j)(i

1UU21U2

1U1U2

U1U

22

2

)(j-F(j)FΔxΔt(j)(i,j),j)(i

(i,j)-F)(i,jFΔxΔt(i,j)(j)

1U()U(2

UU211U

,U()1(UUU

FD Methods

Methods to supress numerical instabilities

Numerical diffusion (first order methods)

2

2UUUx(t,x)D

x(t,x)a

t(t,x)

22

2a

txI

txD

Numerical dispersion

Lax-Wendroff (or second order methods)

Beam-Warming method

Iaxtax 2

2

22

6

3

3UUUx(t,x)

x(t,x)a

t(t,x)

22

22 326

axta

xtIax

Numerical dispersion

( a > 0 ) ( a < 0 )

Numerical dispersion

( a > 0 ) ( a < 0 )

Convergence

Lax-Friedrichs:

Convergence:

xtC (i,j) E

0,0E x (i,j)

FD Methods

+ / Primitive+ / Fast

- / Accuracy- / Numerical instabilities

end

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