Vectorized Parallel in Time methods for low-order discretizations with application to Porous Media problems
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| Format: | Preprint |
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2025
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| author | Engwer, Christian Schell, Alexander Dreier, Nils-Arne |
| author_facet | Engwer, Christian Schell, Alexander Dreier, Nils-Arne |
| contents | High order methods have shown great potential to overcome performance issues of simulations of partial differential equations (PDEs) on modern hardware, still many users stick to low-order, matrix-based simulations, in particular in porous media applications. Heterogeneous coefficients and low regularity of the solution are reasons not to employ high order discretizations. We present a new approach for the simulation of instationary PDEs that allows to partially mitigate the performance problems. By reformulating the original problem we derive a parallel in time time integrator that increases the arithmetic intensity and introduces additional structure into the problem. By this it helps accelerate matrix-based simulations on modern hardware architectures. Based on a system for multiple time steps we will formulate a matrix equation that can be solved using vectorized solvers like Block Krylov methods. The structure of this approach makes it applicable for a wide range of linear and nonlinear problems. In our numerical experiments we present some first results for three different PDEs, a linear convection-diffusion equation, a nonlinear diffusion-reaction equation and a realistic example based on the Richards' equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_02117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vectorized Parallel in Time methods for low-order discretizations with application to Porous Media problems Engwer, Christian Schell, Alexander Dreier, Nils-Arne Numerical Analysis Computational Engineering, Finance, and Science Mathematical Software 86-08, 65J10, 65J15, 65M20, 65Y05, 65Y10, 49M15, 35K61 High order methods have shown great potential to overcome performance issues of simulations of partial differential equations (PDEs) on modern hardware, still many users stick to low-order, matrix-based simulations, in particular in porous media applications. Heterogeneous coefficients and low regularity of the solution are reasons not to employ high order discretizations. We present a new approach for the simulation of instationary PDEs that allows to partially mitigate the performance problems. By reformulating the original problem we derive a parallel in time time integrator that increases the arithmetic intensity and introduces additional structure into the problem. By this it helps accelerate matrix-based simulations on modern hardware architectures. Based on a system for multiple time steps we will formulate a matrix equation that can be solved using vectorized solvers like Block Krylov methods. The structure of this approach makes it applicable for a wide range of linear and nonlinear problems. In our numerical experiments we present some first results for three different PDEs, a linear convection-diffusion equation, a nonlinear diffusion-reaction equation and a realistic example based on the Richards' equation. |
| title | Vectorized Parallel in Time methods for low-order discretizations with application to Porous Media problems |
| topic | Numerical Analysis Computational Engineering, Finance, and Science Mathematical Software 86-08, 65J10, 65J15, 65M20, 65Y05, 65Y10, 49M15, 35K61 |
| url | https://arxiv.org/abs/2504.02117 |