Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting

Abstract

Radiation hydrodynamics are a challenging multiscale and multiphysics set of equations. To capture the relevant physics of interest, one typically must time step on the hydrodynamics timescale, making explicit integration the obvious choice. On the other hand, the coupled radiation equations have a scaling such that implicit integration is effectively necessary in non-relativistic regimes. A first-order Lie-Trotter-like operator split is the most common time integration scheme used in practice, alternating between an explicit hydrodynamics step and an implicit radiation solve and energy deposition step. However, such a scheme is limited to first-order accuracy, and nonlinear coupling between the radiation and hydrodynamics equations makes a more general additive partitioning of the equations non-trivial. Here, we develop a new formulation and partitioning of radiation hydrodynamics with gray diffusion that allows us to apply (linearly) implicitexplicit Runge-Kutta time integration schemes. We prove conservation of total energy in the new framework, and demonstrate 2nd-order convergence in time on multiple radiative shock problems, achieving error 3-5 orders of magnitude smaller than the first-order Lie-Trotter operator split at the hydrodynamic CFL, even when Lie-Trotter applies a 3rd-order TVD Runge-Kutta scheme to the hydrodynamics equations.

Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

“…Another recent work on preconditioning IRK, related to the considered framework, is found in Reference 41. The preconditioner is based on a form of the system matrix, similar to that of , and on a closed form of the inverse of the matrix $$$ {A}_q^{-1}\otimes {I}_n+\tau {I}_q\otimes {M}^{-1}K $$$.…”

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2D Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

“…Moreover, although Gauss and other fully implicit Runge-Kutta methods can offer high accuracy in addition to good conservation properties, historically, high-order methods have rarely been used in practice for numerical PDEs due to the difficulty of solving the fully coupled stage equations. However, recent theoretical and algorithmic developments have made such integration tractable and even faster than diagonally implicit Runge-Kutta (DIRK) methods at times [43,44].…”

Efficient and Parallel Solution of High-order Continuous Time Galerkin for Dissipative and Wave Propagation Problems

“…The difficulty in using the high-order continuous time Galerkin method or any implicit time Runge-Kutta methods is that a straightforward implementation requires to solve a system of linear equations of the size rM × rM , which is not feasible in most time for PDE problems. In a recent work [23], efficient iterative algorithms are developed based on optimal preconditioning of the stage matrix for finding the stage vectors of the implicit Runge-Kutta methods for solving (1.1). For an r-stage implicit Runge-Kutta method, the stage matrix is an r × r block matrix with each block being a M × M matrix.…”

“…For an r-stage implicit Runge-Kutta method, the stage matrix is an r × r block matrix with each block being a M × M matrix. One can find further references in [23] for developing efficient algorithms implementing the high order implicit time discretization methods in the literature. We also refer to [22], [19] for the implementation of the discontinuous time Galerkin method based on the block diagnalization of the stiffness matrix.…”

“…In this paper we propose an efficient realization of the method (1.4) which uses Legendre polynomials as shape functions to obtain a new stiffness matrix which is different from the stage matrix in [23] applying to the Gauss collocation method. By exploiting the special structure of the stiffness matrix, we construct an algorithm which computes the solution Y r (t n ) at each node by solving k complex matrix problems and r − 2k real matrix problems in parallel, where k, 0 ≤ k ≤ r/2, is the number of complex zeros of the [r/r] Padé numerator P r (z).…”