Spectral collocation and waveform relaxation methods with Gegenbauer reconstruction for nonlinear conservation laws

Transcription

1 Spectral collocation and waveform relaxation methods with Gegenbauer reconstruction for nonlinear conservation laws Z. Jackiewicz and B. Zubik Kowal November 2, 2004 Abstract. We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear conservation laws. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated by highly stable implicit methods. The obtained numerical solution is then enhanced on the intervals of smoothness by Gegenbauer reconstruction. The effectiveness of this approach is illustrated by numerical experiments. Keywords: Nonlinear conservation law, pseudospectral methods, waveform relaxation iterations, Gegenbauer reconstruction. Department of Mathematics, Arizona State University, Tempe, Arizona 85287, jackiewi@math.la.asu.edu. The work of this author was partially supported by the National Science Foundation under grant NSF DMS Department of Mathematics, Boise State University, 1910 University Drive, Boise, Idaho 83725, zubik@diamond.boisestate.edu. 1

2 1 Introduction It is the purpose of this paper to investigate spectral collocation and waveform relaxation methods for the nonlinear conservation law u(x, t) + t x f ( u(x, t) ) = 0, L x L, t, u(x, ) = g(x), L x L, (1.1) u( L, t) = α(t), u(l, t) = β(t), t. The solution u(x, t) to (1.1) is a limit of u ν (x, t) as ν 0, where u ν (x, t) satisfies the boundary value problem with added viscosity ν t uν (x, t) + x f ( u ν (x, t) ) = ν 2 x (x, t), L x L, t, u ν (x, ) = g(x), L x L, u ν ( L, t) = α(t), u ν (L, t) = β(t), t, (1.2) and it may be beneficial to consider (1.2) with small viscosity ν instead of (1.2) to stabilize the resulting schemes. Spectral and Legendre pseudospectral viscosity methods for nonlinear conservation laws have been considered by Gottlieb, Lustman and Orszag [17], Maday, Ould Kaber and Tadmor [32], and Guo, Ma and Tadmor [24]. In the next section we will describe the Chebyshev pseudospectral methods for (1.1) and (1.2). This leads to the nonlinear system of ordinary differential equations (ODEs) which will be integrated in time by explicit Runge Kutta methods of order four. The resulting numerical schemes are spectrally accurate if the solution to (1.1) or (1.2) is smooth. For discontinuous problems we can improve the convergence away from the discontinuity by applying a filter of order p. High order filters allow for high resolution away from the discontinuities with strong oscillations (Gibbs phenomenon) in the neighborhoods of the discontinuities. The application of low order filters will reduce these oscillations, but at the expense of excessive blurring. In this paper we will use the former approach (high order filters) with subsequent application of Gegenbauer reconstruction, as in [24], to resolve the Gibbs phenomenon and enhance the accuracy of the resulting numerical approximations. This will be described in Section 3. In Section 4 the 2

3 results of numerical experiments are presented which illustrate the effectiveness of pseudospectral Chebyshev methods with Gegenbauer reconstruction for discontinuous problems. In Section 5 we will discuss waveform relaxation methods for (1.1) and (1.2) in conjunction with Chebyshev pseudospectral methods. These methods can be viewed as a way to replace the nonlinear systems of differential equations obtained by semidiscretization in space by the method of lines of (1.1) or (1.2) by a sequence of linear problems which are easier to solve. As a result, we can employ implicit numerical schemes for integration in time. These methods have much better stability properties than the explicit formulas employed in Section 2 and, as a consequence, allow the use of much larger stepsizes for time integration which fulfill stability restrictions, as compared with the explicit Runge Kutta schemes. This is confirmed by numerical experiments presented in Section 6. The additional advantage of waveform relaxation is the fact that depending on the choice of splitting we can decouple the resulting system of differential equations which can then be efficiently integrated in a parallel computing environment. Finally, in Section 7 some concluding remarks are given and plans for future research are briefly outlined. 2 Pseudospectral Chebyshev method We will describe the pseudospectral Chebyshev method for (1.2), where to simplify the notation, we will skip the dependence of the solution to this problem on the parameter ν. Let M be a nonnegative integer and denote by x i = L cos(πi/m), i = 0, 1,..., M, the Chebyshev Gauss Lobatto points in the interval [ L, L]. We discretize (1.2) in space by the method of lines replacing 2 f(u(x, t)) and u(x, t) by pseudospectral approximations given x x 2 by and i = 0, 1,..., M. Here, x f ( u(x i, t) ) f ( u(x i, t) ) M 2 x 2 u(x i, t) M j=0 j=0 D (k) = [ d (k) ] M ij, k = 1, 2, i,j=0 3 d (1) ij u(x j, t), (2.1) d (2) ij u(x j, t), (2.2)

4 are differentiation matrices of order k. The explicit expressions for these matrices are given in [7], [10] and [38]. Put u i (t) = u(x i, t). Substituting (2.1) and (2.2) into (1.2) and taking into account that u 0 (t) = α(t), u M (t) = β(t), we obtain M 1 u i(t) + f (u i (t)) d (1) M 1 ij u j (t) = ν d (2) ij u j (t) j=1 j=1 f (u i (t)) ( d (1) i0 α(t) + d (1) imβ(t) ) + ν ( d (2) i0 α(t) + d (2) imβ(t) ), (2.3) u i ( ) = g(x i ), i = 1, 2,..., M 1. Put and D (k) = [ d (k) ] M 1 ij, i,j=1 a(k) = u(t) = [ u 1 (t) u M 1 (t) d (k) 1,0. d (k) M 1,0, b(k) = ] T, g = [ f (u(t)) = [ f (u 1 (t)) f (u M 1 (t)) d (k) 1,M. d (k) M 1,M g(x 1 ) g(x M 1 ) ] T., k = 1, 2, ] T, Then the system (2.3) can be rewritten in the following vector form u (t) = f (u(t)) D (1) u(t) + ν D (2) u(t) f (u(t)) (a (1) α(t) + b (1) β(t) ) + ν ( a (2) α(t) + b (2) β(t) ), (2.4) u( ) = g, t, where stands for componentwise multiplication. This system is then integrated in time by the classical Runge Kutta method of order four, compare [6]. If the initial condition g(x) or the solution to (1.1) or (1.2) is discontinuous it may be necessary to use filters to stabilize the resulting scheme and to enhance the accuracy of numerical approximations. The use of filters for discontinuous problems is discussed in [39], [17], [23], [1], [15], [36], and 4

5 [16]. If the initial function g(x) is discontinuous we can obtain the filtered approximation M gm(x σ i ) = fijg(x σ j ), i = 0, 1,..., M, where F σ = [ f σ ij f σ ij = w j ] M M n=0 i,j=0 j=0 is the filter matrix defined by σ ( n ) T n (x i )T n (x j ). M γ n Here, T n (x) = cos(n arccos(x)), n = 0, 1,..., are Chebyshev polynomials, π, j = 0, M, 2M π, n = 0, M, w j = γ π n = M, j = 1, 2,..., M 1, π, n = 1, 2,..., M 1, 2 are the weights of the Chebyshev Gauss Lobatto quadrature formula and the normalization constants, respectively, and σ(η) is a filter of order p, i.e., the real function σ C p 1 (, ) which has the following properties: σ(η) = 0 for η > 1; σ(0) = 1; σ(1) = 0; and σ (m) (0) = σ (m) (1) = 0 for m = 1, 2,..., p 1, compare [39], [15], [16]. The frequently used filters are the Cesáro, raised cosine, sharpened raised cosine, Lanczos, and exponential cut off filter, see [39], [7], [15], [16]. The family of exponential filters of order p is defined by 1, η η c, σ(η) = ( exp α ( η η ) ) c p, η > η c, 1 η c where α is a measure of how strong the various modes should be filtered. Usually α is chosen so that σ(1) is of the order of the machine accuracy eps of the computer. This leads to α = ln(c eps) where C is a constant of moderate size, and we have chosen C = 1 in our implementation of this filter. We have found this family of filters to be quite effective in our numerical experiments reported in Section 4 and 6. We can also obtain matrices which combine the effect of filtering and differentiation of order k. These matrices D (k),σ = [ d (k),σ ] M ij are defined by d (k),σ ij = w j M n=0 σ ( n ) T n (x j )T n (k) (x i ), M γ n 5 i,j=0

6 compare again [15], [16]. The general system of differential equations resulting from the semidiscretization of (1.2) in space using filtering and differentiation matrices D (k),σ takes the form u (t) = f (u(t)) D (1),σ u(t) + ν D (2),σ u(t) f (u(t)) (a (1),σ α(t) + b (1),σ β(t) ) + ν ( a (2),σ α(t) + b (2),σ β(t) ), (2.5) u( ) = F σ g, ] M 1 t, where F σ = [ fij σ, and D (k),σ, a (k),σ, and b (k),σ are defined similarly i,j=1 as D (k), a (k), and b (k). Observe that for σ(η) 1 the system (2.5) reduces to (2.4). It is also possible to use different functions σ(η) for D (k),σ, k = 1, 2, and F σ, but it seems to be difficult to decide which strategy leads to the optimal results. We experimented with and compared many choices and results of some numerical experiments are presented in Sections 4 and 6. We have introduced small viscosity ν in (1.2) and (2.5) in the physical space. Guo, Ma and Tadmor [24] consider alternative approach, the so called spectral vanishing viscosity method, which is implemented directly on high modes of the computed solution in the spectral domain. 3 Location of discontinuities in the solution and Gegenbauer reconstruction After the approximation u h (x i, t end ) to the solution u(x i, t end ) of (1.1) or (1.2) is computed we can compute spectral coefficients a n (t end ), i.e., the coefficients of the expansion of u h (x i, t end ) in terms of Chebyshev polynomials T n (x) u h (x i, t end ) = M n=0 i = 0, 1,..., M, from the formula listed in [17] a n (t end ) = 2 M c n M j=0 a n (t end )T n ( x i ), (3.1) L 1 c j u h (x M j, t end ) cos πjn M, (3.2) where c 0 = c M = 2 and c n = 1 for n = 1, 2,..., M 1. By examination of these coefficients we can then find the number S and the location of discontinuities in the solution u to (2.5). This can be done in many different 6

7 ways and we have adopted the approach by Gottlieb, Lustman and Orszag described in [17]. In this approach the Chebyshev coefficients a n (t end ) (3.2) are fitted by an expression of the form S a n (t end ) = B s A n (X s ) s=1 for M/3 n 2M/3, which corresponds to the spectral representation with coefficients A n (X s ) of the sum of the Heaviside functions with discontinuities at X s and discontinuity jumps B s. As explained in [17] the values of X s are related to the eigenvalues of some summation operator and can be determined independently of B s. This leads to the exact determination of subintervals on which discontinuities are located and they are then assigned to the midpoints of these subintervals. As reported in [17] the number of shocks does not have to be specified in advance and this procedure works quite well for as many as seven shocks. Once X s are computed the values of B s can be found, if needed, by a least squares procedure. We refer to [17] for a complete description of this process. Another approach advocated by Gelb and Tadmor [13] in the context where Fourier spectral coefficients are given, is based on generalized conjugate partial sums whose convergence to the jump function is then accelerated by the so called concentration factors. This approach was generalized in [14] to general Jacobi polynomials which include Chebyshev polynomials as special case. This results in effective edge detectors, where both the location of discontinuities and discontinuity jumps are recovered. We refer again to [13], [14] for details. After the locations of discontinuities are detected we will use next Gegenbauer reconstruction as described in [19], [20], [21] to improve the accuracy of the numerical solution on the subintervals of [ L, L] where the solution to (1.1) or (1.2) is smooth. This reconstruction is based on ultraspherical or Genenbauer polynomials Cn(x), λ i.e., the polynomials which are orthonormal with respect to the inner product C λ k, C λ n = 1 h λ n 1 1 where, for λ > 0, h λ n is defined by (1 x 2 ) λ 1 2 C λ k (x)c λ n(x) dx h λ n = π 1 2 C λ Γ(λ n (1) ) Γ(λ)(n + λ), 7

8 and normalized so that C λ n(1) = Γ(n + 2λ) n! Γ(2λ). They can be computed from the three term recurrence relation (n + 1)C λ n+1(x) = 2(n + λ) x C λ n(x) (n + 2λ 1)C λ n 1(x), C λ 0 (x) = 1, C λ 1 (x) = 2λ x, n = 1, 2,..., compare [2], [8]. To illustrate this procedure let us assume that the solution to (1.1) or (1.2) is smooth on the subinterval [a, b] [ L, L], and, as in [19], define the local variable ξ by x = x(ξ) = ɛ ξ+δ, where ɛ = (b a)/2, δ = (b+a)/2, which transfoms the interval [ 1, 1] onto [a, b]. Denote by x i, i = 0, 1,..., M, the Chebyshev Gauss Lobatto points corresponding to [a, b] and compute the Chebyshev partial sum by ũ h ( x i, t end ) = M n=0 ( xi ) a n (t end )T n, (3.3) L i = 0, 1,..., M. We compute next the Gegenbauer series on a subinterval [a, b] by the formula m gm(x, λ t end ) = ĝɛ λ (l)c ( l λ x δ ), (3.4) l=0 ɛ where ĝ λ ɛ (l) are approximations to the Gegenbauer expansion coefficients g λ ɛ (l) based on a subinterval [a, b]. These coefficients are defined by g λ ɛ (l) = 1 h λ l 1 1 (1 ξ 2 ) λ 1 2 C λ l (ξ)ũ h (ɛ ξ + δ, t end ) dξ. The approximations to these coefficients are computed by evaluating the above integral by the Chebyshev Gauss Lobatto quadrature formula. This leads to ĝɛ λ (l) = 1 M w h λ j (1 x 2 j) λ Cl λ ( x j )ũ(ɛ x j + δ, t end ), (3.5) l j=0 where x j are the Chebyshev Gauss Lobatto points corresponding to the interval [ 1, 1] and w j are the corresponding weights. We expect that g λ m(x, t end ) 8

9 will be a better approximation to u(x, t end ) than u h (x, t end ) on the intervals of smoothness of u. This expectation is supported by the result of Gottlieb and Shu [19] who proved that Gegenbauer reconstruction m ĝɛ λ (l)cl λ (ξ) l=0 applied to the piecewise smooth function u(x) defined on the interval [ 1, 1] with λ = m = β ɛ M, β < 2/27, is uniformly convergent to u(x) on the interval of smoothness [a, b] of u as M. To be more precise, the resulting error can be bounded by max 1 ξ 1 u(ɛ ξ + δ) m l=0 ĝ λ ɛ (l)c λ l (ξ) A ( q ɛ M T ) + qr ɛ M, where ( ) 27β β q T = < 1, qr = 2 ( ) 27ɛ β < 1, 32ρ and A grows at most as M 1+2λ. Here, ρ is the distance from [a, b] to the nearest singularity of u(x) in the complex plane. The constants q T and q R correspond to truncation and regularization errors, see [19], [20], [21]. The implementation of Gegenbauer reconstruction is quite sensitive to round off errors and it is not clear how to choose m and λ to obtain optimal results. Gottlied and Shu [19] recommend the values and Guo, Ma, and Tadmor [24] choose m = 0.1 ɛ M, λ = 0.2 ɛ M, (3.6) m = λ = 0.05 M, but they did not attempt to optimize these parameters. To our knowledge the first attempt to optimize these parameters was made by Gelb [11] for Gegenbauer reconstruction where the Fourier coefficients are given for a smooth but non periodic function. This approach was further refined in [12] by taking into account the smoothness characteristics of the function. The determination of optimal parameters based on the Chebyshev spectral coefficients is discussed in [25]. 9

10 M=20 M=20 M=40 M= M=80 M=80 error M= x Figure 1: Errors of the Genenbauer reconstruction with m and λ defined by (3.6) for M = 20, 40, 80, and 160. To illustrate the effectiveness of Gegenbauer reconstruction we have plotted in Fig. 1 the error of this procedure for the function f(x) given by exp(x), x [ 1, 0], f(x) = sin(cos(x)) + 1, x (0, 1], with the parameters m and λ defined by (3.6) for M = 20, 40, 80, and 160. The performance of this procedure could be further improved by appropriate tuning of the parameters m and λ, compare again [11], [12], [25]. The graphs of errors versus M in double logarithmic scale which illustrate spectral convergence of Gegenbauer reconstruction method for functions with varying degrees of smoothness are presented in [25] and [12]. 10

11 4 Numerical experiments with pseudospectral methods and Gegenbauer reconstruction In this section we present the results of some numerical experiments for the Burgers equation u(x, t) + u(x, t) t x u(x, t) = 0, L x L, t [, t end ], u(x, ) = x, x < 2A, u(x, ) = 0, x > 2A, (4.1) u(±l, t) = ± L t, t [, t end ], where A > 0 is a parameter. The solution to this problem is the N wave x u(x, t) = t, x < 2At, 0, x > 2At, compare [40]. This solution is the limit as ν 0 of the solutions to the problem u(x, t) + u(x, t) t x 2 u(x, t) = ν u(x, t), (4.2) x2 L x L, t [, t end ], with appropriate initial and boundary conditions. The solution to (4.2) is given by u(x, t) = x ( t e x2/(4νt) ) 1 1 +, t e A/(2ν) 1 compare again [40]. The pseudospectral Chebyshev method applied to (4.2) leads to the system of ODEs u (t) = u(t) D (1),σ u(t) + ν D (2),σ u(t) u(t) (a (1),σ α(t) + b (1),σ β(t) ) + ν ( a (2),σ α(t) + b (2),σ β(t) ), (4.3) u( ) = F σ g, 11

12 x x M Figure 2: Numerical results for M = 64 (left graphs) and error versus M (right graph). t, where g correspond to initial function at the grid points x i and F σ, D (k),σ, a (k),σ, and b (k),σ are defined in Section 2. We have solved the problem (4.3) with = 1, t end = 2 and A = 1 by pseudospectral Chebyshev method (2.5). This system was integrated by the classical Runge Kutta method of order four. The efficiency of numerical computations could be increased somewhat by the use of total variation diminishing (TVD) or strong stability preserving (SSP) Runge Kutta methods (compare [22], [35]). However, time integration is not the main focus of this paper and we do not pursue these issues in more detail here. After integrating (2.5) in time, we then computed Chebyshev spectral coefficients (3.2) which were used to locate the discontinuities in the solution 12

13 and for Gegenbauer reconstruction as described in Section 3. In Fig. 2 (left graphs) we present the results for M = 64, ν = 1/M, time step t = 10 3, D (1),σ = D (1), D (2),σ = D (2), where we filter the initial condition using F σ corresponding to the exponential filter with p = 4 and η c = 2/M. The parameters used for Gegenbauer reconstruction were chosen by trial and error and they are m = 1 and λ = 4 for left and right subintervals and m = 2 and λ = 5 for the middle subinterval. In the left upper graph of this figure the exact solution is plotted by thick solid line, the numerical solution before postprocessing and the not a knot spline fit of it by black square and thin solid line and the numerical solution obtained by Gegenbauer reconstruction by white circle. In the left lower graph we have plotted the not a knot spline fit of the error before postprocessing by dashed line and the error after postprocessing by solid line. These global errors are defined as norms of the differences between the exact solution to (4.2) and the corresponding numerical solutions. We can see that Gegenbauer postprocessing is very effective in reconstructing the solution to (4.1) on the subintervals of smoothness. In Fig. 2 (right graph) we have plotted in double logarithmic scale error versus M (black circles connected by thin solid line) together with the spline fit (thick solid line) generated using program csaps.m from Matlab Spline Toolbox (Version 3.2.1). This graph illustrates spectral convergence of the overall numerical scheme. The parameters m and λ of Gegenbauer reconstruction were chosen again by trial and error for different values of M and different subintervals. The systematic approach to choosing these parameters is the subject of current research, compare [12]. 5 Waveform relaxation methods Waveform relaxation method is an iterative technique to compute successive approximations u (k) (t), k = 0, 1,..., to the solution u(t) of (2.5). To be more specific, we use previous iterate u (k) (t) in the argument of the function f and the next iterate u (k+1) (t) elsewhere. This has the effect of replacing the 13

14 nonlinear system (2.5) by a sequence of linear problems of the form d dt u(k+1) (t) = f (u (k) (t)) D (1),σ u (k+1) (t) + ν D (2),σ u (k+1) (t) f (u (k) (t)) (a (1),σ α(t) + b (1),σ β(t) ) + ν ( a (2),σ α(t) + b (2),σ β(t) ), (5.1) u (k+1) ( ) = F σ g, t, k = 0, 1,.... We can also consider more general schemes corresponding to appropriate splittings of matrices D (l),σ, i.e., D (l),σ = (l),σ (l),σ D 1 + D 2, l = 1, 2. For example, choosing Gauss Jacobi or block Gauss Jacobi splittings D (l),σ ) (l),σ (l),σ 1 = diag ( D 1,1,..., D 1,s, l = 1, 2, (l),σ (l),σ where D 1,j, j = 1, 2,..., s, are diagonal blocks of D 1 of dimension m j, m m s = M 1, leads to the sequence of linear problems d dt u(k+1) (t) = f (u (k) (1),σ (t)) ( D 1 u (k+1) (1),σ (t) + D 2 u (k) (t) ) (2),σ + ν ( D 1 u (k+1) (2),σ (t) + D 2 u (k) (t) ) (5.2) f (u (k) (t)) (a (1),σ α(t) + b (1),σ β(t) ) + ν ( a (2),σ α(t) + b (2),σ β(t) ), u (k+1) ( ) = F σ g, t, k = 0, 1,..., where u (0) (t) is an arbitrary starting function usually chosen as u (0) (t) = F σ (l),σ g, t. Observe that the form of the matrices D effectively decouples the resulting system (5.2) into s subsystems which can be assigned to different processors for efficient solution in a parallel computing environment. For any function u : [, T ] R M 1 define the supremum norm on the interval [, T ] by u [t0,t ] = sup{ u(t) : t [, T ]}, where is any norm in R M 1. To study boundedness of the sequence u (k) defined by (5.2) we assume that f (u) F (Q) for u Q, (5.3) 14

15 where F (Q) is a constant depending on Q. We also introduce the following notation: µ i (T ) = a (i),σ α + b (i),σ β [t0,t ], i = 1, 2, and C i = C i (Q, ν, σ) = F (Q) D (1),σ i + ν D (2),σ i, i = 1, 2, C 3 = C 3 (Q, ν, T ) = F (Q)µ 1 (T ) + νµ 2 (T ). We can demonstrate that the sequence u (k) defined by (5.2) is uniformly bounded for sufficiently small values of T. To be more precise we have the following theorem. Theorem 5.1 Assume that the function f satisfies (5.3) and the starting function u (0) satisfies u (0) [t0,t ] Q, for some constant Q 2 F σ g + 1. Then k = 0, 1,..., if T is chosen so that u (k) [t0,t ] Q, (5.4) (T )C 1 < 1 2 and (T )(C 2 Q + C 3 ) < 1 2. (5.5) Proof: The result will follow if we can show that (5.4) is satisfied for k + 1 assuming that it is satisfied for k. Integrating (5.2) from to T and taking norms in the resulting equation and using (5.3) we obtain or u (k+1) [t0,t ] F σ g + (T ) ( C 1 u (k+1) [t0,t ] + C 2 u (k) [t0,t ] + C 3 ), u (k+1) [t0,t ] F σ g + (T )(C 2 u (k) [t0,t ] + C 3 ) 1 (T )C 1. Since, by assumption, (5.4) is satisfied for k if T satisfies (5.5) it follows that u (k+1) [t0,t ] 2 ( F σ g + (T )(C 2 Q + C 3 ) ) 2 F σ g + 1 Q which proves (5.4) for k

16 To study the convergence of the sequence u (k) defined by (5.2) in the uniform norm we assume that f (u 1 ) f (u 2 ) L u 1 u 2, (5.6) for some constant L 0. We also introduce the notation ( C 4 = C 4 (L, Q, σ, T ) = L Q ( D (1),σ 1 + D (1),σ 2 ) ) + µ 1 (T ). Put e (k) (t) = u(t) u (k) (t), k = 0, 1,..., where u(t) is the solution to (2.5) and u (k) (t) are defined by (5.2). We have the following theorem. Theorem 5.2 Assume that the function f satisfies (5.3) and (5.6) and that T satisfies (5.5). Then e (k) (t) e C 1(t ) (C 5(t )) k E 0, (5.7) k! k = 0, 1,..., t [, T ], where C 5 = C 2 + C 4 and E 0 is some constant such that e (0) [t0,t ] E 0. Proof: Subtracting (5.2) from (2.5), then integrating from to t and taking norms on both sides of the resulting equation and using (5.3), (5.4) and (5.6) after some computations we obtain e (k+1) (t) C 1 k = 0, 1,..., t [, T ]. recurrence relation ɛ (0) (t) = e (0) (t), k = 0, 1,..., t [, T ]. inequalities that e (k+1) (s) ds + C 5 e (k) (s) ds, Define the sequence of functions ɛ (k) (t) by the ɛ (k+1) (t) = C 1 The equation (5.8) is equivalent to ɛ (k+1) (s)ds + C 5 ɛ (k) (s)ds, (5.8) Then it follows from the theory of differential e (k) (t) ɛ (k) (t). d dt ɛ(k+1) (t) C 1 ɛ (k+1) (t) = C 5 ɛ (k) (t), t [, T ], ɛ (k+1) ( ) = 0, 16

17 which has the solution e C 1(t ) ɛ (k+1) (t) = C 5 Define the sequence ɛ (k) (t) by e C 1(s ) ɛ (k) (s)ds. ɛ (k) (t) = e C 1(t ) ɛ (k) (t). Then ɛ (k) (t) satisfies the recurrence equation ɛ (k+1) (t) = C 5 k = 0, 1,..., which has the solution ɛ (k) (t) = C k 5 ɛ (k) (s)ds, (t s) k 1 (k 1)! ɛ(0) (s)ds. (5.9) To see that this is the case observe that this relation is satisfied for k = 1. Assuming that (5.9) is satisfied for k we obtain ɛ (k+1) (t) = C k+1 5 = C k+1 5 = C k+1 5 s ξ (t ξ) k k! (s ξ) k 1 ɛ (0) (ξ) dξ ds (k 1)! (s ξ) k 1 (k 1)! ɛ (0) (ξ) dξ ɛ (0) (ξ) ds dξ which proves (5.9) for k + 1. Since ɛ (0) (t) = e C 1(t ) ɛ (0) (t) E 0, (5.9) implies that ɛ (k) (t) C k 5 (t s) k 1 (k 1)! ds E 0 = (C 5(t )) k k! E 0, (5.10) k = 0, 1,..., t [, T ], which proves (5.7). The estimate (5.10) proves the superlinear convergence of the sequence u (k) to the solution u of (4.2). This estimate can be improved by using the relation ɛ (k) (t) = C5 k (t s) k 1 E 0 (k 1)! ec 1(t s) ds 17

18 instead of (5.9). This leads to the error bound e (k) (t) Ck 5 E 0 (k 1)! t0 0 s k 1 e C 1s ds, k = 0, 1,..., expressed in terms of the integral of the function s k 1 exp(c 1 s), which is a sharper estimate than (5.7). We can also show that the sequence u (k) is uniformly bounded on arbitrarily long intervals [, T ]. To do so we assume that Q > F σ g + 1 and use the exponential norm where α is chosen so that u α [,T ] := sup { u(t) e α(t ) : t [, T ] }, (5.11) α > max { C 1, C 3 /( F σ g + 1), (C 1 + C 2 )Q/(Q F σ g 1) }. We have the following theorem. Theorem 5.3 Assume that the function f satisfies (5.3) and that the starting function u (0) satisfies with Q > F σ g. Then for k = 0, 1,.... u (0) (t) Q, t [, T ] (5.12) u (k) α [,T ] Q (5.13) Proof: Note that (5.12) implies (5.13) with k = 0. Assume (5.13) for k. Integrating (5.2) from to t, then taking norms on both sides of the resulting equation and using (5.3) we obtain u (k+1) (t) F σ g + C 1 + C 2 u (k+1) (s) e α(s ) e α(s ) ds u (k) (s) e α(s ) e α(s ) ds + C 3 F σ g + C 1 u (k+1) α [,T ] + C 2 u (k) α [,T ] e α(s ) ds e α(s ) ds + C 3 (t ) F σ g + C 1 α u(k+1) α [,T ] e α(t ) + C 2 α u(k) α [,T ] e α(t ) + C 3 (t ). 18 ds

19 Hence, u (k+1) (t) e α(t ) ( F σ g + C 3 (t ) ) e α(t ) Observe that since α > C 3 ( F σ g + 1) we have + C 1 α u(k+1) α [,T ] + C 2 α u(k) α [,T ]. ( F σ g + C 3 (t ) ) e α (t ) F σ g + 1 and it follows that This leads to α u (k+1) α [,T ] α ( F σ g + 1) + C 1 u (k+1) α,t ] + C 2 Q. u (k+1) α [,T ] α ( F σ g + 1) + C 2 Q α C 1 Q. The last inequality follows from the definition of α. This completes the proof. Finally, we study the convergence of u (k) to the solution u of (2.5) in the exponential norm (5.11) on arbitrary interval [, T ]. Define the constants Q = Qe α(t ) C i = C i ( Q, (1),σ ν, σ) = F ( Q) D i + ν D (2),σ i, i = 1, 2, C 3 = C 3 ( Q, ν, T ) = F ( Q)µ 1 (T ) + νµ 2 (T ), C 4 = C 4 (L, Q, ( σ, T ) = L Q ( D (1),σ 1 + D (1),σ 2 ) ) + µ 1 (T ). We have the following theorem. Theorem 5.4 Assume that the function f satisfies (5.3) and (5.6). Then e (k) α [,t] e C 1 (t ) ( C 5 (t )) k Ẽ 0, (5.14) k! k = 0, 1,..., t [, T ], where C 5 = C 2 + C 4 and Ẽ0 is some constant such that e (0) α [,T ] Ẽ0. 19

20 Proof: Observe that u k (t) α [,T ] Q implies that uk (t) Q. Subtracting (5.2) from (2.5) and using (5.13) and (5.3) with Q replaced by Q after some computations we obtain e (k+1) (t) C 1 + C 5 k = 0, 1,..., t [, T ]. Hence, e (k+1) α [,t] C 1 C 1 e α(t ) + C 5 e α(t ) e (k+1) (s) e α(s ) e α(s ) ds e (k) (s) e α(s ) e α(s ) ds e (k+1) α [,s] ds e (k) α [,s] ds, e (k+1) α [,s] ds + C 5 e (k) α [,s] ds. The theorem now follows using exactly the same arguments as in the proof of Theorem 5.2. Similarly as before the error bound in Theorem 5.4 can be replaced by e (k) α [,t] C 5k Ẽ0 t0 s k 1 e C 1 s ds, (k 1)! 0 k = 0, 1,..., expressed in terms of the integral of the function s k 1 exp( C 1 s), which is a sharper estimate than (5.14). 6 Numerical experiments with waveform relaxation methods and Gegenbauer reconstruction In this section we will use again Burgers equations (4.1) and (4.2) as test problems for waveform relaxation. The sequence of linear systems of ODEs 20

21 corresponding to (5.2) takes now the form d dt u(k+1) (t) = u (k) (1),σ (t) ( D 1 u (k+1) (1),σ (t) + D 2 u (k) (t) ) (2),σ + ν ( D 1 u (k+1) (2),σ (t) + D 2 u (k) (t) ) u (k) (t) (a (1),σ α(t) + b (1),σ β(t) ) + ν ( a (2),σ α(t) + b (2),σ β(t) ), (6.1) u (k+1) ( ) = F σ g, t, k = 1, 2,.... To integrate these systems in time we apply the implicit x x M Figure 3: Numerical results after three iterations of Gauss Jacobi waveform relaxation method with M = 64. Runge Kutta methods of order four based on the Gauss Legendre quadra- 21

22 ture formula [6]. This method can be represented by Butcher tableaux (6.2) We have solved problem (5.2) with = 1, t end = 2 and A = 1 by waveform relaxation method corresponding to the Gauss Jacobi splitting of the matrices D (l),σ, l = 1, 2. We then computed Chebyshev spectral coefficients (3.2) which were used to locate the discontinuities in the solution and for Gegenbauer reconstruction as described in Section 3. In Fig. 3 (left graphs) we present the results of numerical experiments after three iterations, i.e., solving the system (6.1) for k = 0, 1 and 2, for M = 64, ν = 1/M and time step t = 10 2 which is an order of magnitude larger than that used in Section 4 (l),σ (l),σ for explicit Runge Kutta of order 4. The matrices D 1, D 2 and F σ correspond to the exponential filter with p = 4 and η c = 2/M. The parameters used for Gegenbauer reconstruction are m = 1 and λ = 3. As in Section 4 on the upper graph of this figure the exact solution is plotted by thick solid line, the numerical solution before postprocessing and the not a knot spline fit of it by black square and thin solid line and the numerical solution obtained by Gegenbauer reconstruction by white circle. In the lower graph we have plotted the not a knot spline fit of the error before postprocessing by dashed line and the error after postprocessing by solid line. These errors are defined as in Section 4. In Fig. 3 (right graph) we have plotted in double logarithmic scale error versus M (black circles connected by thin solid line) together with the spline fit (thick solid line) generated using program csaps.m from Matlab Spline Toolbox (Version 3.2.1). This graph illustrates spectral convergence of the overall numerical scheme. As in Section 4 the parameters m and λ of Gegenbauer reconstruction were chosen again by trial and error for different values of M and different subintervals. Similar results are obtained if BDF3, the backward differentiation method of order three, is used with the same time step for integrating the system (6.1). 22

23 7 Concluding remarks We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear conservation laws. The location of discontinuities in the solution is determined by fitting the spectral coefficients corresponding to the numerical solution by spectral representation of a sum of Heaviside functions, and the numerical solution is then enhanced on the intervals of smoothness by Gegenbauer reconstruction. The systems of differential equations resulting from application of spectral collocation methods are solved by explicit Runge Kutta method of order four which require rather small time steps for stable integration. The main advantage of waveform relaxation consists of the fact that we can replace these nonlinear systems of ODEs by a sequence of linear problems which can then be effectively integrated by A stable implicit Runge Kutta methods or A(α) stable backward differentiation methods. This allows for much larger time steps than those used for explicit methods. This is confirmed by numerical experiments presented in Sections 4 and 6. Another advantage of waveform relaxation is that it can be applied in a parallel computing environment. Future work will address the numerical solution of nonlinear conservation laws and Gegenbauer reconstruction in two or three space dimensions, and the implementation of waveform relaxation methods with adaptive window control strategy. Some progress for Gegenbauer reconstruction in two dimension for rectangular domains using tensor product approach has been reported in [30]. The theoretical background for adaptive window strategy was formulated in [5]. Acknowledgements. The authors wish to express their gratidude to the anonymous referees for their useful comments. References [1] S. Abarbanel, D. Gottlieb and E. Tadmor, Spectral methods for discontinuous problems. In: Numerical Methods for Fluid Dynamics II. Proceedings of the 1985 Conference on Numerical Methods in Fluid Dynamics (K.W. Morton and M.J. Baines, eds.), pp , Clarendon Press, Oxford

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