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A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

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Abstract

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.

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Notes

  1. For the (4,4) scheme, \(\theta = \frac{1}{12}\) and thus \(Kh_x^2 = -\frac{1}{\theta CFL^2}<-12\) whenever \(CFL<1\), which is already guaranteed by the stability condition (see Sect. 3).

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Steven Britt & Eli Turkel

  2. Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC, 27695, USA

    Semyon Tsynkov

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141700

    Semyon Tsynkov

Authors
  1. Steven Britt
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  2. Eli Turkel
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  3. Semyon Tsynkov
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Corresponding author

Correspondence to Steven Britt.

Additional information

Work supported by the US-Israel Binational Science Foundation (BSF) under Grant # 2014048 and by US Army Research Office (ARO) under Grant # W911NF-16-1-0115. S. Britt was supported by the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel Aviv University and a Fulbright Postdoctoral Scholarship funded by the US-Israel Educational Foundation.

Numerical Study of Iterative Methods for the Modified Helmholtz Equation

Numerical Study of Iterative Methods for the Modified Helmholtz Equation

In the course of our study of iterative methods for the wave equation, some observations were made regarding the use of MG and CG for solving the modified Helmholtz equation.

Fig. 3
figure 3

For the 2nd order central difference scheme (left), Jacobi without damping leads to divergence for the Poisson equation, which is well known. The 4th order compact scheme (right) exhibits convergence within a small number of V-cycles even without damping for the Poisson equation

Full size image

It is well known that the number of V-cycles needed for the Poisson equation when using the second-order central difference stencil grows with the grid size when using a Jacobi smoother without damping (i.e., \(\omega = 1\)) and it is therefore advantageous to seek a damping parameter for the high frequencies. In Fig. 3, we confirm this classical result for the 2nd order central difference scheme with \(\omega = 1\); however the compact scheme converges with \(\omega = 1\) in a small number of V-cycles. The test solution is \(u = \sin 5x\sin 3y\) on a square domain of side length \(s = \pi \) centered at the origin and Dirichlet boundary conditions on all edges. Each V-cycle uses \(\nu _1 = \nu _2 = 4\) pre- and post-sweeps of the Jacobi smoother. For the second order central difference scheme, a Jacobi smoother with \(\omega = 1\) was also divergent for the modified Helmholtz equation with \(K<0\) but converged rapidly with \(\omega = 4/5\), the classical optimal value for the Poisson equation in 2D. By contrast, use of the optimal damping parameter \(\omega ^*\) (see (45), Sect. 4.2) conferred no advantage for the 4th order compact scheme either for the Poisson or modified Helmholtz equation with \(K<0\).

In Sect. 4.1, our analysis showed that the error bound (38) for conjugate gradient was only well behaved when \(Kh_x^2\) does not tend to zero as the grid is refined. The wave equation resulted in favorable cases in which the modified Helmholtz equation satisfied \(Kh_x^2 = \frac{1}{\theta CFL^2}\), and this quantity is constant for the (4,4) scheme and actually increasing with the grid size for the (2,4) scheme with \(CFL = h_x\). In the following example, we solve the modified Helmholtz equation with a fixed parameter \(K = -50\) using the test solution \(u = \sin {15x}\sin {13y}\) on a square of side length 2 centered at the origin and Dirichlet BCs. The residual tolerance for terminating CG iterations is \(10^{-10}\) in Table 14, while the number of MG V-cycles is the point at which the residual converges.

Table 14 The number of CG iterations increases for the modified Helmholtz equation with a fixed parameter \(K = -50\) while the number of MG V-cycles remains constant
Full size table

Table 14 shows that the number of CG iterations doubles as the grid is refined by a factor of 2 for the case when K is fixed while the number of MG V-cycles remains small. This indicates that MG will in general be more efficient than CG when solving the modified Helmholtz equation. We ran a set of computations for various constant values of \(K=k^2\) and found that the error was a function of kh, indicating that there is no pollution effect for the modified Helmholtz equation.

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Britt, S., Turkel, E. & Tsynkov, S. A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound. J Sci Comput 76, 777–811 (2018). https://doi.org/10.1007/s10915-017-0639-9

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