Efficient time integration for discontinuous Galerkin approximations of linear wave equations

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro-magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves.

[1]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[2]  P. M. van den Berg,et al.  A modified Lanczos algorithm for the computation of transient electromagnetic wavefields , 1997 .

[3]  Marlis Hochbruck,et al.  Preconditioning Lanczos Approximations to the Matrix Exponential , 2005, SIAM J. Sci. Comput..

[4]  Marlis Hochbruck,et al.  Residual, Restarting, and Richardson Iteration for the Matrix Exponential , 2010, SIAM J. Sci. Comput..

[5]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

[6]  Arne Taube,et al.  A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations , 2009 .

[7]  Christian Wieners,et al.  Distributed Point Objects. A New Concept for Parallel Finite Elements , 2005 .

[8]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[9]  Volker Grimm,et al.  Convergence Analysis of an Extended Krylov Subspace Method for the Approximation of Operator Functions in Exponential Integrators , 2013, SIAM J. Numer. Anal..

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

[11]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[12]  Stefan Güttel,et al.  Deflated Restarting for Matrix Functions , 2011, SIAM J. Matrix Anal. Appl..

[13]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[14]  Thomas Bohlen,et al.  On the propagation characteristics of tunnel surface‐waves for seismic prediction , 2010 .

[15]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[16]  N. Higham.  Functions Of Matrices , 2008 .

[17]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[18]  S. Güttel.  Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection , 2013 .

[19]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[20]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[21]  Christian Wieners,et al.  A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing , 2010, Comput. Vis. Sci..

[22]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[23]  Yousef Saad,et al.  Efficient Solution of Parabolic Equations by Krylov Approximation Methods , 1992, SIAM J. Sci. Comput..

[24]  Rob Remis,et al.  A Krylov Stability-Corrected Coordinate-Stretching Method to Simulate Wave Propagation in Unbounded Domains , 2012, SIAM J. Sci. Comput..

[25]  Miguel A. Fernández,et al.  Explicit Runge-Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems , 2010, SIAM J. Numer. Anal..

[26]  L. Fezoui,et al.  Convergence and stability of a discontinuous galerkin time-domain method for the 3D heterogeneous maxwell equations on unstructured meshes , 2005 .

[27]  Jaap J. W. van der Vegt,et al.  Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations , 2007, J. Sci. Comput..

[28]  Davit Harutyunyan,et al.  The Gautschi Time Stepping Scheme for Edge Finite Element Discretization of the Maxwell Equations , 2005 .

[29]  Philippe Villedieu,et al.  Convergence of an explicit finite volume scheme for first order symmetric systems , 2003, Numerische Mathematik.

[30]  Christian Wieners,et al.  A parallel block LU decomposition method for distributed finite element matrices , 2011, Parallel Comput..

[31]  Marcus J. Grote,et al.  Explicit local time-stepping methods for Maxwell's equations , 2010, J. Comput. Appl. Math..

[32]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[33]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[34]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms , 2006 .

[35]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[36]  Vladimir Druskin,et al.  Solution of time-convolutionary Maxwell's equations using parameter-dependent Krylov subspace reduction , 2010, J. Comput. Phys..

[37]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

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

[39]  R. Nagel,et al.  A Short Course on Operator Semigroups , 2006 .

[40]  Fernando Reitich,et al.  High-order RKDG Methods for Computational Electromagnetics , 2005, J. Sci. Comput..

[41]  Lilia Krivodonova,et al.  An efficient local time-stepping scheme for solution of nonlinear conservation laws , 2010, J. Comput. Phys..

[42]  Volker Grimm.  Resolvent Krylov subspace approximation to operator functions , 2012 .

[43]  Marlis Hochbruck,et al.  Implicit Runge-Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations , 2015, SIAM J. Numer. Anal..