An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

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.

[1]  Weiwei Sun,et al.  Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations , 2007, SIAM J. Numer. Anal..

[2]  Nicolas Crouseilles,et al.  An exponential integrator for the drift-kinetic model , 2017, Comput. Phys. Commun..

[3]  William Gropp,et al.  A hybrid format for better performance of sparse matrix-vector multiplication on a GPU , 2016, Int. J. High Perform. Comput. Appl..

[4]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[5]  Lawrence M. Murray,et al.  GPU Acceleration of Runge-Kutta Integrators , 2012, IEEE Transactions on Parallel and Distributed Systems.

[6]  Colm Clancy,et al.  On the use of exponential time integration methods in atmospheric models , 2013 .

[7]  L. Reichel.  Newton interpolation at Leja points , 1990 .

[8]  Lukas Einkemmer,et al.  Semi-Lagrangian Vlasov simulation on GPUs , 2019, Comput. Phys. Commun..

[9]  Terry Haut,et al.  Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems , 2018, Int. J. High Perform. Comput. Appl..

[10]  Alexander Ostermann,et al.  Magnus integrators on multicore CPUs and GPUs , 2017, Comput. Phys. Commun..

[11]  Jürg Nievergelt,et al.  Parallel methods for integrating ordinary differential equations , 1964, CACM.

[12]  Margaret H. Wright,et al.  The opportunities and challenges of exascale computing , 2010 .

[13]  Martin J. Gander,et al.  A Direct Solver for Time Parallelization , 2016 .

[14]  Martin J. Gander,et al.  ParaDIAG: Parallel-in-Time Algorithms Based on the Diagonalization Technique , 2020, ArXiv.

[15]  Alexander Ostermann,et al.  Exponential integrators on graphic processing units , 2013, 2013 International Conference on High Performance Computing & Simulation (HPCS).

[16]  Xu Guo,et al.  Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs , 2013, Comput. Vis. Sci..

[17]  Andrew J. Wathen,et al.  Preconditioning and Iterative Solution of All-at-Once Systems for Evolutionary Partial Differential Equations , 2018, SIAM J. Sci. Comput..

[18]  L. Einkemmer,et al.  Exponential methods for solving hyperbolic problems with application to kinetic equations , 2019, J. Comput. Phys..

[19]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[20]  Matt Martineau,et al.  GPU-STREAM v2.0: Benchmarking the Achievable Memory Bandwidth of Many-Core Processors Across Diverse Parallel Programming Models , 2016, ISC Workshops.

[21]  Gregory Beylkin,et al.  Near optimal rational approximations of large data sets , 2013 .

[22]  Nathan Paldor,et al.  An invariant theory of the linearized shallow water equations with rotation and its application to a sphere and a plane , 2011 .

[23]  Lukas Einkemmer,et al.  Evaluation of the Intel Xeon Phi 7120 and NVIDIA K80 as accelerators for two-dimensional panel codes , 2015, PloS one.

[24]  Martin J. Gander,et al.  50 Years of Time Parallel Time Integration , 2015 .

[25]  Awad H. Al-Mohy,et al.  Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators , 2011, SIAM J. Sci. Comput..

[26]  Marco Caliari,et al.  The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential , 2015, SIAM J. Sci. Comput..

[27]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[28]  Vu Thai Luan,et al.  Parallel exponential Rosenbrock methods , 2016, Comput. Math. Appl..

[29]  Chiara Piazzola,et al.  A splitting approach for the magnetic Schrödinger equation , 2016, J. Comput. Appl. Math..

[30]  Per-Gunnar Martinsson,et al.  A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator , 2016 .

[31]  Sebastian Reich,et al.  Linearly implicit time stepping methods for numerical weather prediction , 2006 .

[32]  Juan Sánchez,et al.  Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells , 2014, J. Comput. Phys..

[33]  V. Maz'ya,et al.  On approximate approximations using Gaussian kernels , 1996 .

[34]  Ian W. Turner,et al.  GPU Accelerated Algorithms for Computing Matrix Function Vector Products with Applications to Exponential Integrators and Fractional Diffusion , 2016, SIAM J. Sci. Comput..