Abstract
Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.
Highlights
We are interested in simulating linear partial differential equations (PDEs) with purely imaginary eigenvalues of large modulus
The original Rational exponential integrators (REXI) scheme is already a good method to compute the action of the matrix exponential parallel in time
3.6 s 0.27 s 7.2 s 0.55 s 239 s 19.35 s 3.4 s 0.47 s 6.5 s 0.91 s 31.0 s 4.13 s 18.4 s 4.14 s 47.6 s 10.16 s 95.6 s 20.57 s proposed a modification of the REXI approach that achieves accuracy close to machine precision at similar or, for some problems, even lower computational cost
Summary
We are interested in simulating linear partial differential equations (PDEs) with purely imaginary eigenvalues of large modulus (i.e. stiff problems). Even exponential integrators of high order, which are able to perform large time steps, are mainly implemented sequentially (in time). It is worth mentioning that REXI is markedly distinct from time parallelization schemes such as the parareal and similar methods (see, e.g., [19]) In the latter case, a coarse and a fine time integrator are combined to achieve parallelism within an iterative procedure, while in the former the action of the matrix exponential is directly approximated in a way that is amendable to parallelization. The proposed method allows us to determine how many terms the approximation requires in order to obtain accurate results We demonstrate an implementation on modern GPUs that yields a drastic speedup compared to the corresponding CPU implementation for the shallow water equations (section 4)
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