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New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids. The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations.
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This content will become publicly available on June 30, 2026
Third-Order Sectorially A-Stable Alternating Implicit Runge–Kutta Schemes
We design pairs of six-stage, third-order, alternating implicit Runge–Kutta (RK) schemes that can be used to integrate in time two stiff operators by an operator-splitting technique. We also design for each pair a companion explicit RK scheme to be used for a third, nonstiff oper- ator in an implicit-explicit (IMEX) fashion. The main application we have in mind is (non)linear parabolic problems, where the two stiff operators represent diffusion processes (for instance, in two spatial directions) and the nonstiff operator represents (non)linear transport. We identify necessary conditions for linear sectorial A( )-stability by considering a scalar ODE with two (complex) ei- genvalues lying in some fixed cone of the half-complex plane with nonpositive real part. We show numerically that it is possible to achieve A(0)-stability when combining two operators with negative eigenvalues, irrespective of their relative magnitude. Finally, we show by numerical examples includ- ing two-dimensional nonlinear transport problems discretized in space using finite elements that the proposed schemes behave well.
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- Award ID(s):
- 2110868
- PAR ID:
- 10640471
- Publisher / Repository:
- SIAM J. SCI. COMPUT. © 2025 Society for Industrial and Applied Mathematics Vol. 47, No. 3, pp. A1579–A1603
- Date Published:
- Journal Name:
- SIAM Journal on Scientific Computing
- Volume:
- 47
- Issue:
- 3
- ISSN:
- 1064-8275
- Page Range / eLocation ID:
- A1579 to A1603
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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