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The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization at each time step. The scheme is fully explicit and relies on the computation of matrix exponential vector products. To accelerate such computation, we further construct a noniterative, nonoverlapping domain decomposition algorithm, namely localized ETDRK2, which loosely decouples the system at each time step via suitable interface conditions. Temporal error analysis of the proposed global and localized ETDRK2 schemes is rigorously proved; moreover, the schemes are shown to be conservative under periodic boundary conditions. Numerical results for the Burgers' equation in one and two dimensions (with moving shocks) are presented to verify the theoretical results and illustrate the performance of the global and localized ETDRK2 methods where large time step sizes can be used without affecting numerical stability.
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On the stability of explicit finite difference methods for advection–diffusion equations
Abstract In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.
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- PAR ID:
- 10444035
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 39
- Issue:
- 1
- ISSN:
- 0749-159X
- Page Range / eLocation ID:
- p. 421-446
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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