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Abstract In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step.
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This content will become publicly available on June 1, 2026
Noniterative localized exponential time differencing methods for hyperbolic conservation laws
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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- Award ID(s):
- 2041884
- PAR ID:
- 10620756
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Advances in Computational Mathematics
- Volume:
- 51
- Issue:
- 3
- ISSN:
- 1019-7168
- Subject(s) / Keyword(s):
- Exponential time differencing Runge-Kutta discontinuous Galerkin nonoverlapping domain decomposition hyperbolic conservation laws temporal error estimates.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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