HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms
HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms
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DOI：
10.1007/978-3-030-39647-3_9
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发表时间：
2020
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影响因子：
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通讯作者：
T. Babb;P. Martinsson;Daniel Appelö
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文献类型：
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作者：
T. Babb;P. Martinsson;Daniel Appelö
本章描述了一个计算效率很高的求解器，它的形式是κ∂u∂t= Lu (x, t)+ h (u, x, t), x∈Ω， t> 0,(1)，初始数据为u (x, 0)= u0 (x)。这里L是作用于固定域Ω的椭圆算子，h是低阶的，可能是非线性的项。我们取κ为实数或虚数，允许抛物线型和Schrödinger型方程。我们希望从隐式求解器中获得好处，例如l稳定性和刚性精度，这意味着计算瓶颈将是在Ω上设置的一系列椭圆方程的解。在每个时间步要求解的椭圆方程相同的情况下，使用直接（而不是迭代）求解器是非常有利的。在直接求解器中，构造了椭圆方程的近似解算子。构建它的成本通常高于使用多重网格等迭代方法求解单个椭圆所需的成本，但好处是，在构建之后，每个后续求解都非常快。在本章中，我们认为在这种情况下使用的特别有效的直接求解器是通过结合多域谱搭配离散化(a
In this chapter describes a highly computationally efficient solver for equations of the form κ∂ u∂ t= Lu (x, t)+ h (u, x, t), x∈ Ω, t> 0,(1) with initial data u (x, 0)= u0 (x). Here L is an elliptic operator acting on a fixed domain Ω and h is lower order, possibly nonlinear terms. We take κ to be real or imaginary, allowing for parabolic and Schrödinger type equations. We desire the benefits that can be gained from an implicit solver, such as L-stability and stiff accuracy, which means that the computational bottleneck will be the solution of a sequence of elliptic equations set on Ω. In situations where the elliptic equation to be solved is the same in each time-step, it is highly advantageous to use a direct (as opposed to iterative) solver. In a direct solver, an approximate solution operator to the elliptic equation is built once. The cost to build it is typically higher than the cost required for a single elliptic solve using an iterative method such as multigrid, but the upside is that after it has been built, each subsequent solve is very fast. In this chapter, we argue that a particularly efficient direct solver to use in this context is a method obtained by combining a multidomain spectral collocation discretization (a