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Summation-by-parts (SBP) finite difference methods are widely used in scientific applications alongside a special treatment of boundary conditions through the simultaneous-approximate-term (SAT) technique which enables the valuable proof of numerical stability. Our work is motivated by multi-scale earthquake cycle simulations described by partial differential equations (PDEs) whose discretizations lead to huge systems of equations and often rely on iterative schemes and parallel implementations to make the nu- merical solutions tractable. In this study, we consider 2D, variable coefficient elliptic PDEs in complex geometries discretized with the SBP-SAT method. The multigrid method is a well-known, efficient solver or preconditioner for traditional numerical discretizations, but they have not been well-developed for SBP-SAT methods on HPC platforms. We propose a custom geometric-multigrid pre- conditioned conjugate-gradient (MGCG) method that applies SBP- preserving interpolations. We then present novel, matrix-free GPU kernels designed specifically for SBP operators whose differences from traditional methods make this task nontrivial but that perform 3× faster than SpMV while requiring only a fraction of memory. The matrix-free GPU implementation of our MGCG method per- forms 5× faster than the SpMV counterpart for the largest problems considered (67 million degrees of freedom). When compared to off- the-shelf solvers in the state-of-the-art libraries PETSc and AmgX, our implementation achieves superior performance in both itera- tions and overall runtime. The method presented in this work offers an attractive solver for simulations using the SBP-SAT method.
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An Energy-Based Summation-by-Parts Finite Difference Method For the Wave Equation in Second Order Form
Abstract We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method.
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- PAR ID:
- 10328446
- Date Published:
- Journal Name:
- Journal of Scientific Computing
- Volume:
- 91
- Issue:
- 2
- ISSN:
- 0885-7474
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s10915-022-01829-4
