Parallel-in-Time Solution of Hyperbolic PDE Systems via Characteristic-Variable Block Preconditioning
- Univ. of Waterloo, ON (Canada); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Univ. of Waterloo, ON (Canada)
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Univ. of New Mexico, Albuquerque, NM (United States)
We consider the parallel-in-time solution of both linear and nonlinear hyperbolic partial differential equation (PDE) systems in one spatial dimension. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that intervariable coupling between characteristic variables is weak, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small, while that between the original variables is not. For an ℓ-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of ℓ scalar linear(ized)-advection-like problems, each associated with a different characteristic wave-speed in the underlying linear(ized) PDE. Furthermore, we approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions. For many test problems, the solver converges in just a handful of iterations and with mesh-independent convergence rates.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- Grant/Contract Number:
- AC52-07NA27344; 89233218CNA000001
- OSTI ID:
- 3001304
- Report Number(s):
- LA-UR--25-20668; LLNL--JRNL-2000224; 10.1137/24M1673310
- Journal Information:
- SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing; ISSN 1064-8275; ISSN 1095-7197
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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