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<p>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 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 <i>linear convection-diffusion equation</i>, a <i>nonlinear diffusion-reaction equation</i> and a realistic example based on the <i>Richards’ equation</i>.</p>

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Vectorized Parallel in Time methods for low-order discretizations with application to Porous Media problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月5日

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  • Christian Engwer,
  • Alexander Schell,
  • Nils-Arne Dreier

摘要

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 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.