Journal of Scientific Computing, Vol. 12, No. I, 1997

Polynomial Time-Marching for NonreflectingBoundary Problems

Yong Luo1 and Matthew J. Yedlin'

Ren-ited April 12, 1995

The newly developed polynomial time-marching technique has been successfullyextended to nonperiodic boundary condition cases. In this paper, a special non-periodic boundary condition, nonreflecting or absorbing boundary condition, isincorporated into the pseudospectral polynomial time-marching scheme. Thus,this accurate and stable time-dependent PDE solver can be applied to someopen domain or free space problems. The balanced overall spectral accuracy isillustrated by some numerical experiments in the one-dimensional case. Theerror goes to zero at a rate faster than many fixed orders of the finite-differencescheme. The order of the absorbing boundary approximation is addressed inone-dimension. Also, in the two-dimensional case, a 2nd-order absorbingapproximation has been incorporated into the polynomial time-marchingscheme with Chebyshev collocation in space. Comparison with the previousfinite-difference implementation indicates that the high stability and efficiency ofthe polynomial time-marching remains. The overall accuracy is mainly limitedby the 2nd-order absorbing approximation.

1. INTRODUCTION

When simulating wave propagation, it is necessary to introduce artificialboundaries to limit the area of computation, due to the finite memorylimitation of the computer. The boundary conditions at these artificialboundaries are used to guarantee a unique and well-posed solution to thePartial Differential Equation (PDE) problems within the computationaldomain. These artificial boundary conditions are expected to affect the

1 Department of Electrical Engineering, University of British Columbia, Vancouver, BritishColumbia, V6T IZ4 Canada. E-mail: {yongl, matty}a iee.ubc.ca.

31

0885-7474 '97 0300-0031$ I 2.500 1997 Plenum Publishing Corporation

KEY WORDS: Absorbing boundary condition; nonreflecting boundary condi-tion; time-marching.

solution in a manner such that it closely approximates the free space solu-tion which exists in the absence of these boundaries. Thus, the amplitudesof waves reflected from these artificial boundaries are expected to be mini-mized.

In order to avoid (or reduce) the edge reflection contamination fromthe computational domain boundaries, one may make the model suf-ficiently large so that the arrival times of these edge reflections are outsidethe time window of interest. Although this method results in a wavefieldfree of any edge contamination, this option is very costly in terms of CPUtime and memory. Another more practical choice is the implementation ofnonreflecting and/or absorbing boundary conditions at the computationaldomain edges. The term "nonreflecting" here means that a set of equationsare imposed only at those grid points on the edges of the computa-tional domain that mathematically absorb almost all the outgoing energy[(Engquist and Majda, (1977), Renaut, (1992)]. Unfortunately, accordingto Engquist and Majda, (1977), the perfectly absorbing conditionsnecessarily have to be nonlocal in both space and time and thus are notuseful for practical calculations. Hence, in practice, some approximationshave to be derived to approach the theoretical nonlocal boundary condi-tion, at the cost of some energy being reflected into the computationaldomain. Another alternative is the use of an absorbing region to dampenthe outgoing energy by surrounding the main domain with a narrowdamping strip which can drastically dampen the traversing waves. Thedamping factor should go from zero in the interior of the strip to a maxi-mum value at the edge of the whole model. This technique is more efficientthan the large "free-space" one, but still involves nontrivial extra wavepropagation. The third alternative to eliminate the edge reflection is to setsome boundary conditions which can be used to cancel the edge reflectioncompletely [(Goode, (1993)]. As described by Goode, (1993), Dirichletand Neumann boundary conditions can be used to simulate rigid and freesurface conditions respectively. Their combination can be used to effectivelyand completely cancel the first order edge reflections. Unlike otherschemes, it is independent of the incident angle. However, in order toobtain the combined solutions of both boundary conditions, 2' wavepropagation problems (i is the number of boundaries involved), one foreach combination of Dirichlet and Neumann conditions applied, have tobe solved and linearly superimposed. Therefore, this perfectly-absorbingscheme is also very costly in computing time, although it's simpler.

In Section 2 of this paper, we will briefly review the general approxi-mation methods for the nonreflecting boundary conditions. The poly-nomial time-marching scheme for nonperiodic boundary problems will beaddressed in Section 3. Some numerical experiments have been described in

32 Luo and Yedlin

Section 4 to show the fast error convergence in the balanced accuracy ofthis time-marching scheme. In Section 5, the method of incorporation ofthe exact absorbing boundary conditions into the new polynomial time-marching scheme will be described in detail. In Section 6, the eigenvaluebound calculation of the two-dimensional operator will be discussed asbackground for the polynomial time-marching scheme with the two-dimen-sional absorbing boundaries. Finally, some numerical examples will bepresented in Section 7 to illustrate that this incorporation does not usecomputer resources which are significantly greater than the fixed boundaryconditions such as Dirichlet or Neumann. Unlike the finite-differencescheme, which requires half-cell information and a different differencingscheme at the boundary cells, the polynomial time-marching can directlyincorporate the 2nd-order absorbing boundary approximation into thespatial operator without involving additional computing or manipulations.Some numerical examples, in one and two dimensions, will also be presentedto demonstrate the accuracy and the efficiency of the new scheme.

2. NONREFLECTION BOUNDARY CONDITIONAPPROXIMATION

Let us consider the scalar wave equation,

If the Fourier transform of u(x, y, t) over y and t is given by u(x, & w),then p(d/dy, d/dt) can be represented as p ( , w ) , a smooth function homo-geneous of degree zero for || + \w\ large with support in ,2 > w2 for (,, w)large and identically one on a neighborhood of the support of u(0, , w)(Fourier transform of u(x, y, t) at x = 0).

Unfortunately, the perfectly absorbing boundary condition given in(2.2) is necessarily nonlocal in both space and time. This boundary conditionis impractical from a computational point of view since the advance of onetime level at a single point requires information from all previous timesover the entire boundary. Therefore, many highly absorbing local approxi-mations to (2.2) have been developed. Necessarily, the boundary conditionapproximations need to satisfy the following two criteria [Engquist andMajda, (1977)]:

854/12/1-3

Nonreflecting Boundary Problems 33

According to Engquist and Majda, (1977), the perfectly absorbing bound-ary condition for this equation at x = 0 can be written in the form

34 Luo and Yedlin

1. These boundary conditions are local. This is essential forreasonable control of the operation count.

2. The boundary conditions lead to a well-posed mixed boundaryvalue problem for the wave equation. This condition guarantees thestability of the solution.

The first and second order approximations based on the Taylor orFade expansions are [Engquist and Majda, (1977].

1st Order Approximation:

2nd Order Approximation:

Either of these approximations should be solved at the boundary x = 0 (orthe lower x bound of the domain). Both of these two approximations havebeen verified to be well-posed in [Engquist and Majda, (1977), Trefethenand Halpern, (1986)]. The first order in (2.3) one approximates theabsorption at normal incidence and is perfect in the one-dimensional spacecase. Actually, the condition in (2.3) is the standard left-traveling one-waywave equation, which represents only the left-traveling wave at the bound-ary x = 0.

Renaut (1992) reviewed some of the higher order methods currentlyused for solving the two-dimensional scalar wave equations with absorbingboundary conditions, such as Lindman (1975), Engquist and Majda,(1977). Renaut (1992) showed that the extension to higher orders is neitherimmediately obvious nor unique, as there are many different ways one candiscretize the derived absorbing boundary condition. She also proposeda new high-order absorbing boundary condition approximation, whichemploys the 2nd order space derivative as the highest order derivative inthe formula. For the two-dimensional scalar wave equation,

the absorbing boundary condition at x = 0 can be approximated by a setof equations (as opposed to one)

Nonreflecting Boundary Problems 35

which must be solved at the boundary. Here p0, a.,,Bi are approximationcoefficients. In this equation, to increase the degree of approximation, oneonly needs to add some new coefficients and increase the loop index (i.e.,increase the degree of interpolation, but not the highest order derivative).This property s quite useful in the finite-difference implementation.

In the polynomial time-marching scheme, due to the complicatedmatrix manipulations, only the standard 2nd-order approximation in (2.4)will be considered as an example for the absorbing boundary conditionimplementation in this next time-marching scheme.

3. POLYNOMIAL TIME-MARCHING WITH NONPERIODICBOUNDARY CONDITIONS

A standard pseudospectral scheme for solving time-dependent differen-tial equations is to approximate the space derivatives by spectral collocationand to march the solution in time by a finite-difference approach. Usingconventional finite-difference methods for the temporal discretizationresults only in finite-order accuracy in time. The error of the fully-discretized method will be dominated by the temporal errors. Tal-Ezer hasdeveloped a time-differencing method which has infinite order accuracy.Tal-Ezer has developed a time-differencing method which has infinite orderaccuracy. The time dependence of the solution is expressed as a series ofChebyshev polynomials which can be generated by a three-term recurrencerelation. As Canuto et al. (1988) commented, this method is more restrictedin applicability but substantially more efficient and accurate when it doesapply. The major limitation of the Tal-Ezer time-marching method is thatit requires a normal spectral spatial operator with all its eigenvalues beingeither purely real or purely imaginary. In practice, this method only appliesto wave equations without diffusion and damping terms. Due to this limita-tion, Tal-Ezer (1989, 1991) extended the polynomial time-marching conceptto a more general Faber polynomial or Newton-form polynomial inter-polation. By this complex matrix function approximation technique, onecan obtain a spectrally accurate time marching scheme for more generalwave equations.

If the spectral basis function includes the boundary behavior, such asperiodic or otherwise, then the spectral operator automatically includes the

36 Luo and Ycdlin

boundary behavior. The Tal-Ezer time-marching technique, which is basedon the approximation of the spectral operator function, can be directlyapplied. The Newton-form polynomial interpolation time-marching techni-que has been successfully developed to nonperiodic cases in Luo and Yedlin,(1994). Consider a Dirichlet initial-boundary-value problem as shown here:

If we use Chebyshev collocation at Gauss-Lobatto points

and if we also set

the symbolic solution can be represented as

where u = [u1,..., U N _ 1 ] and v = [v1,,..., V N _ 1 ] are ( N l ) x l vectors forthe interior grid points, I N _ 1 is an identity matrix of rank (N 1), D2spis a Chebyshev collocation differentiation operator with homogeneousDirichlet boundary conditions. According to Luo and Yedlin, (1994), it isdefined as the 2nd order Chebyshev collocated derivative matrix with itsfirst, last rows and columns deleted respectively.

Similarly, according to Luo and Yedlin, (1994), the symbolic solutionto the similar Neumann initial-boundary-value problem is in the sameform, but now both U = [ U O , . . . , U N ] and v- [v0,..., VN] are (N + l) x lvectors for all the collocation points, and

Nonreflecting Boundary Problems 37

where the D2sp is the Chebyshev collocation differentiation operator withhomogeneous Neumann boundary conditions and is given by

This ( N + l ) x ( N + l ) matrix D2sp implicitly reflects both the Chebyshevcollocated space differentiation at interior Gauss-Lobatto points and thehomogeneous Neumann boundary conditions at end points. All thederivative matrix elements di} are defined in Gottlieb et al, (1984).

Once the symbolic solution and the relevant operator representationhave been obtained, the time-dependence can be approximated by theNewton form polynomial interpolation at the Fejer points: [Tal-Ezer(1989, 1991)]

and

where p is the logarithmic capacity of the eigenvalue distribution domain.The detailed procedure of calculating the interpolation coefficients and theFejer points is introduced and described by Tal-Ezer (1989, 1991). Due toTal-Ezer (1989, 1991), and Luo and Yedlin (1994). The time-marchingscheme based on the Newton form polynomial interpolation results in asubstantial improvement in resolution, accuracy and stability over conven-tional finite difference methods.

4. SPECTRAL ACCURACY DISCUSSION OF THE POLYNOMIALTIME-MARCHING, SCHEME IN ONE DIMENSION

Fornberg (1987) stated the pseudospectral method performs in manysituations far better than present theory would suggest. As for the spectral

38

Table I. Time-Marching Errors for the One-Dimensional Wave Equation with HomogeneousDirichlet Boundary Conditions (Homogeneous

Medium )

N

48

121620

Mean error

1.641 E-25.568 E-51.841E-81.423E-122.806 E- 14

Max. error

4. 102 E-21.050E-46.150E-84.535 E-l 25.684 E- 14

Luo and Ycdlin

accuracy for general initial-boundary value problems, no general theory isavailable that is as readily applicable as that for finite-differences. Toinvestigate the spectral accuracy of the Chebyshev pseudospectral in poly-nomial time-marching scheme, some one-dimensional experimental calcula-tions have been performed to obtain the data in Tables I and II.

For Table I, the Dirichlet case, the computation is based on Eq. (3.1)with U0(x) = sin(nx) for the Dirichlet case. For Table II, the Neumannboundary condition case, the initial distribution is cosine. A time step ofzl? = 0.01 is used. The errors are computed from the 100th time step of thewave distribution, compared with the corresponding analytical solution.From the error data listed in the tables, it is clear that the error convergesto zero rapidly. In the Neumann case (Table II), the mean error ratio ofjV= 8 to N 4 is about 0.0046, which roughly shows an error at the orderof O({1/N)8); i.e., 8th order finite-difference scheme accuracy. The meanerror ratio of N= 16 to 7V=8 is about 8.14e 8, which roughly shows anerror at the order of O((l/N)24); i.e., 24th order finite-difference schemeaccuracy. In the Dirichlet case (Table I), the mean error ratio of TV =8 to

Table II. Time-Marching Errors for the One-Dimensional Wave Equation with HomogeneousNeumann Boundary Conditions (Homogeneous

Medium)

TV

48

121620

Mean error

6.041 E-l2.788 E-31.552E-62.270 E-101.099 E-l 3

Max. error

7.0IOE-13.048 E-31.593E-62.297E-101.634 E-13

Nonreflecting Boundary Problems 39

N = 4 is approximately 0.0034. It also shows an 8th finite-difference orderof error convergence. The mean error ratio of N= 16 to TV =8 is 2.56e 8,which is roughly equivalent to an error O((l/N)25); i.e., 25th order finite-difference accuracy. Both Dirichlet and Neumann case results show thatwhen the number of grid points N increases, the order of error convergencealso increases. Although it is claimed by Boyd (1989) that in the ellipticequation case the spectral method has an error of O(( l/N)N), in this experi-ment, probably due to the boundary condition interaction and also due tothe relatively low accuracy of computing the Schwarz-Christoffel transfor-mation at small ./Vs, the results do not illustrate the kind of "exponentialorder" convergence. In fact, due to the limitation of the computer's doubleprecision format, in practice, it is impossible to obtain this order of con-vergence for N>10. However, the experimental results do show that thedegree of error convergence increases with N.

5. POLYNOMIAL TIME-MATCHING WITH NONREFLECTINGBOUNDARYCONDITIONS

In the polynomial time-marching method, as discussed in Section 3,we need to incorporate the time-dependent absorbing boundary conditionsinto the spatial operator. Here, we can take advantage of the 1st order timederivative term in the vector form in (3.3) and (3.4). Let's consider a one-dimensional scalar wave equation with perfectly absorbing boundaries attwo ends:

Let v = u,. Then if we collocate the data at Gauss-Lobatto points (same asthe manipulations for (3.1)), the boundary conditions can be exactlyrepresented at the end points Xo= 1, XN = +1:

40 Luo and Yedlin

where u0, UN, v0, VN are the collocated data at the end points x0= 1,XN +1. Therefore, the derivatives can be represented as

Nonreflecting Boundary Problems 41

The matrix D1 and the elements dij are defined in Gottlieb et al., (1984). Thevectors H, v are the same as in the previous Section. By the same procedureas described in (3.3) and (3.4), we can obtain the general symbolic operatorsolution including the absorbing boundary condition as shown next:

By a similar method as described in the eigenvalue calculation procedurein Luo and Yedlin, (1994), one can also get the eigenvalue bounds of[GN](2N+2)2 directly from those of [D1, B] and [D1.D1]. Let l be aneigenvalue of the operator [GN~][2N + 2}2, then we have

Here d, g respectively are the eigenvalues of [D1, D1] and [D, B]. This,the maximum eigenvalue (spectral radius) of the operator [ G N ] ( 2 N + 2 ) 2 canbe calculated from the eigenvalues of [D1, D1 ] and [D1, B]. Direct eigen-value computation from the bigger matrix [GN]2N+2f is unnecessary. Thespectral radius of this bigger matrix will be used in setting up the Schwarz-Christoffel transformation, which determines the time-marching polynomial.We have demonstrated how to include the absorbing boundary conditionin the one-dimensional wave operator. Furthermore, we have computed thenecessary spectral radius for the operator.

The next two tables present the error convergence results for theapplication of absorbing boundary condition to 2 cases: (1) Table III:

854 12 1-4

42 Luo and Yedlin

Table HI. Time- Marching Errors for the One-Dimensional Wave Equation with Absorbing

Boundary Conditions (Homogeneous Medium)

N

64128

Mean error

5.629 E-55.374 E-8

Max. error

2.038 E-41.401E-7

a homogeneous medium and (2) Table IV: a layered medium where c = 0.5for -1 < x < 0, c = 1/0 for 0 < x ^ 1 with a transition point at x = 0.

In Table III, the mean error ratio from N = 128 to N = 64 is 9.55e - 4,which shows an error convergence order of O((l/N)l). This degree ofconvergence is approximately the same as that for 10th order of finite-difference. Probably due to the operator complexity in this case, the mini-mum error caused by Schwarz-Christoffel conformal mapping is O(10~9).Thus, the overall accuracy is certainly limited by this order. A time step ofAt = 0.001 is used in this experiment. The errors are computed from thewave at 240th time step. A cosine initial distribution is used.

Finally, a step discontinuity with the absorbing boundary condition isanalyzed to see the effect of the medium variance. The experimental datalisted in Table IV show that in varying medium (wave velocity jumpingfrom 0.5 to 1.0 with one middle point transition) the performance is quitepoor (O(l/N)) due to the effect of a varying wave speed. Severe accuracydegradation is caused by the step discontinuity. In the polynomial time-marching scheme, since the differential operator is applied repeatedly overthe domain, some high order derivatives of the medium variation con-taminate the overall solution. In the computation of Table IV data, thetime step remains as

Nonrcflecting Boundary Problems 43

6. THE TWO-DIMENSIONAL SCALAR WAVE EQUATIONWe now proceed to analyze the case of two-dimensional wave

propagation. In this case, the Chebyshev collocation is done along the Xand Y directions. The eigenvalue analysis for the collocated derivativeoperator is more complicated. Suppose we can represent the two-dimen-sional Laplace operator as here:

Here, D(x2),D(y2) include homogeneous boundary conditions and can beobtained using the procedure described in the previous sections. Instead ofa one-dimensional vector, here U is a two-dimensional matrix with x-row-F-column orientation:

For this orientation, we have the following relationships for the operatormatrices:

Where D'^,,, D'p are the corresponding one-dimensional Chebyshev col-located 2nd order derivative operators (including homogeneous boundaryconditions) for N+l and M+ 1 grid points, respectively.

To get the eigenvalue bounds of the operator L2, we need to useKronecker product and to stack the matrix U(N + 1 ) * ( M + 1 ) for form avector V

44 Luo and Yedlin

The Kronecker product (sometimes called the direct product] is definedas: [Barnett, (1990)]

Each submatrix in this (mp) x (nq) matrix has dimensions p x q. TheKronecker product is distributive and associative [Barnett, (1990)]. Basedan the description of Barnett (1990), the matrix equation

is equivalent to

Here, vec( ) is used to represent the re-stacking procedure along thecolumn, as described in (6.4). Hence, we can re-write (6.1) as:

According to the property of Kronecker product [Barnett, (1990)], if D(2x]has eigenvalues 1,-, i = 0, I,..., N, and D(y,2) has eigenvalues uj = Q, I,..., M,then the Kronecker product operator in (6.8) has (M+l)(N+l) eigen-values lk = ht + u j , k = 0, 1,..., ( M + 1 ) ( N + I). Thus, the bounds of the two-dimensional Chebyshev collocation operator can be directly obtained fromthe maximum bound of the corresponding one-dimensional operatorbounds along X and Y. The eigenvalue bounds of the operator determinethe size and the shape of the domain D, thereby determining thelogarithmic capacity and the distribution of the Fejer points used in (3.7).The multi-dimensional wave operator with absorbing boundary conditionsand the corresponding spectral radius can be obtained using a procedureanalogous to that described in Section 5.

In order to get a good absorbing approximation, one needs to use atleast the 2nd order absorbing boundary condition approximation (2.4),rather than just the 1st order one for the normal incidence. Consider a two-dimensional scalar wave equation within the rectangular domain x, y e[ 1, +1 ]. The boundary condition at the edge of x = 1 is a 2nd orderabsorbing approximation. At the other 3 edges ( x = + l , y=+/ 1),homogeneous Dirichlet conditions are sued. Therefore, we have

Nonreflecting Boundary Problems 45

Discretizing N, M respectively along x, y at Gauss-Lobatto points, wehave

Similarly, if we set v = u1, based on the two-dimensional results in (6.1), wehave the following matrix expression:

where

46 l.iio and Yedlin

Here, d(i,j, d(,i,j2} are respectively the elements of Dt, D2 defined in Gottliebet al., (1984) and di,j = [DT2]i, j. The first row, first and last columns of thesolution matrix uO, j, u i 0 , ui, M are all zero, due to the 3-edge homogeneousDirichlet conditions.

The foregoing results for both one and two dimensions will beemployed in the numerical examples that follow.

7. NUMERICAL EXAMPLESThe Figs. 1 and 2 show some numerical results for the one-dimension

absorbing boundary condition approximation when N =32 and N=64.The initial pulses are all Gaussian, the time step is 0.2 and the wavevelocity c = 1. The time-marching relative error control factor is less than10 ~5. The analytical reference can be derived from the physical interpre-tation of one-dimensional scalar wave propagation. The reference will beused to calculate the error of the absorbing approximation.

Fig. 1. One-dimension absorbing boundary condition numerical results (N = 32).

Nonreflecting Boundary Problems 47

Fig. 2. One-dimension absorbing boundary condition numerical results (N = 64).

The left column of the Figs. 1 and 2 shows the wave amplitude att = 0.4, 1.0, and 1.6 (when the wave pulse completely propagate out of the[ -1, +1] interval), the " + " represents the initial wave pulse. The corre-sponding graphics in the right column illustrate the magnitude of the errorat t = 0.4, 1.0, and 1.6. It is quite clear that the absorbing approximationis perfect because there's no significant change in the magnitude errorbefore and during the absorbing process (the r = 0.4 case can be treated asbefore any actual absorbing process since the wave pulse has not arrivedat the end points yet). Note that at t=l.6 the error and the waveamplitude are identical. An ideal absorber would result in a wave with zeroamplitude at t= 1.6.

For the two-dimensional case, to evaluate the error of the absorbingboundary approximation, another two simulation results need to belinearly superimposed to get a perfectly absorbing edge at x= 1. One caseis with all 4 edges of homogeneous Dirichlet, and the second is with homo-geneous Neumann at the edge of x = 1 but the rest all homogeneous

Dirichlet. According to Goode, (1993), their linear combination willsimulate the ideal absorbing boundary at the edge of x = 1.

In order to get the absorbing effect in a few time steps and also toavoid the reflection interference from the other full-reflection edges, theinitial Gaussian pulse center is located very close to the x = 1 boundary.The initial pulse is tapered to zero at the near-edge elements. The initialpulse is given by:

48 Lno and Yedlin

where the wave velocity c = 1.0 and the time step At = 0.01. If speed of lightunits are used, c = 3 x 108 m/sec and, At = 3.33 x 10~'' second. This is com-parable to 2.5 x 10 ~" second time step used in Tirkas et al, (1992). In thisexperiment, the number of collocation grid points in both X and Y is 32and xQ = 0.5290. That is, the peak of the initial pulse will pass the x = 1boundary at 47th time step. The global error and normalized reflection arealso defined in the same way as in Tirkas et al., (1992). At each time stepthe error is defined to be

The global error is defined as

The normalized reflection is the reflection occurring at X N _ 1 =cos(n(Nl)/N), i.e., one cell next to the XN= 1 absorbing boundary,when the pulse peak passes the boundary. It is normalized by the maxi-mum value of the pulse along the boundary at that moment (47th timestep). The top two diagrams in Fig. 3 show that the global error growswith times (because more wave energy arrives at the boundary as timeincreases). Within 80 time steps, i.e., before the wave pulse arrives the othernonabsorbing boundaries (y = +1), the global error is always very small(compared to the time-marching error control factor 10~5. The bottomdiagram in Fig. 3 demonstrates that the maximum reflection occurring atthe boundary x I, where the absorbing approximation boundary con-dition is applied, is about 5%.

The global errors of both cases (A= 0.015 and A 0.002) in thisexperiment are comparable to those reported in reference [Tirkas et a/.,(1992)] (less than 0.0003 with 80 time steps). When A =0.015, the initialpulse is steeper and more aliasing may occur. Thus, the results are worse(the global error grows a little faster). Since the same polynomial time-marching and Chebyshev collocation scheme are used here for both the

Nonreflecting Boundary Problems 49

Fig. 3. Two-dimension absorbing approximation numerical results (grid points 32 in both Xand Y, dt = 0.01).

approximated and the perfect absorber, the global error only demonstratesthe accuracy of the absorbing approximation, which is limited by the orderof the absorbing boundary condition's accuracy, as predicted in Section 4.

8. CONCLUSIONSAnalysis and experiments in this paper have illustrated the higher

overall accuracy of the polynomial time-marching scheme. With numericalboundary conditions or perfect absorbers, the error can converge at a ratefaster than many fixed orders of the finite-difference scheme. Also, it hasbeen shown that the accurate polynomial time-marching combined withthe Chebyshev collocation can be extended to included the 2nd-orderabsorbing boundary condition approximation without involving anysignificant changes or adding additional computation. The 2nd-orderabsorbing approximation can also be represented with spectral accuracy2nd-order absorbing approximation can also be represented with spectralaccuracy at the boundary cells, in contrast to the finite-difference schemes,

where the absorbing formulas are always approximated by finite differencesin time and space with finite-order accuracy. Since the absorber accuracy islimited by the 2nd-order Taylor expansion, the inherit error in the numericalresults shown in dominated by the analytical 2nd order approximation toan ideal absorber. The implementation of the approximate absorber isspectrally accurate, whereas the FD approximation is finite order (2nd or4th) accurate. Of course, no numerical scheme will eliminate the effect ofusing an approximate absorber. In general, this polynomial time-marchingscheme is superior in stability, resolution and temporal polynomial time-marching scheme is superior in stability, resolution and temporal accuracyas discussed in Tal-Ezer, (1989, 1991); and Luo and Yedlin, (1994).

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difference time-domain method, IEEE Trans, on Ant. & Prop. 40(10), 1215-1222.Trefethen, L, and Halpern, L. (1986). Well-posedness of one-way wave equations and absorbing

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50 Luo and Yedlin