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.

翻译：高阶方法已展现出克服现代硬件上偏微分方程（PDE）模拟性能瓶颈的巨大潜力，然而许多用户仍坚持使用低阶、基于矩阵的模拟方法，尤其是在多孔介质应用中。异质系数和解的低正则性是避免采用高阶离散化的原因。我们提出了一种模拟非稳态PDE的新方法，能够部分缓解性能问题。通过对原问题进行重构，我们推导出一种并行时间积分器，该积分器提高了算术强度并为问题引入了额外的结构。这有助于在现代硬件架构上加速基于矩阵的模拟。基于一个多时间步系统，我们将构建一个矩阵方程，该方程可使用向量化求解器（如块Krylov方法）进行求解。此方法的结构使其适用于广泛的线性和非线性问题。在数值实验中，我们针对三个不同的PDE给出了初步结果：一个线性对流-扩散方程、一个非线性扩散-反应方程以及一个基于Richards方程的实际案例。