Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs (original) (raw)
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Order‐Optimal Preconditioners for Implicit Runge–Kutta Schemes Applied to Parabolic PDEs
SIAM Journal on Scientific Computing, 2007
In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank-Nicolson schemes can be reused for higher-order fully implicit Runge-Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order-optimal for A-stable and irreducible Runge-Kutta schemes with invertible coefficient matrices. The theoretical investigations are confirmed by numerical experiments.
A New Block Preconditioner for Implicit Runge--Kutta Methods for Parabolic PDE Problems
SIAM Journal on Scientific Computing, 2021
A new preconditioner based on a block LDU factorization with algebraic multigrid subsolves for scalability is introduced for the large, structured systems appearing in implicit Runge-Kutta time integration of parabolic partial differential equations. This preconditioner is compared in condition number and eigenvalue distribution, and in numerical experiments with others in the literature: block Jacobi, block Gauss-Seidel, and the optimized block Gauss-Seidel method of [5]. Experiments are run with implicit Runge-Kutta stages up to s = 7, and it is found that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.
SIAM Journal on Scientific Computing
Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to diagonally implicit RK (or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by block-circulant preconditioners and Jordan form solution techniques for IRK. We then derive an efficient, near-optimal singly-diagonal approximate BRSD (SABRSD) by approximating the quasi-triangular matrix in real Schur decomposition using an optimized upper-triangular matrix with a single diagonal value. A desirable feature of SABRSD is that it has comparable memory requirements and factorization (or setup) cost as singly DIRK (SDIRK). We approximate the diagonal blocks in these preconditioning techniques using an incomplete factorization with (near) linear complexity, such as multilevel ILU, ILU(0), or a multigrid method with an ILU-based smoother. We apply the block preconditioners in right-preconditioned GMRES to solve the advection-diffusion equation in 3D using finite element and finite difference methods. We show that BCSD, BRSD, and BJF significantly outperform other preconditioners in terms of GMRES iterations, and SABRSD is competitive with them and the prior state of the art in terms of computational cost while requiring the least amount of memory.
2022
Comp. v. 43, pp. 475-495) on order-optimal preconditioners for parabolic PDEs to a larger class of differential equations and methods. The problems considered are those of the forms ut = −Ku + g and utt = −Ku + g, where the operator K is defined by Ku := −∇ • (α∇u) + βu and the functions α and β are restricted so that α > 0, and β ≥ 0. The methods considered are A-stable implicit Runge-Kutta methods for the parabolic equation and implicit Runge-Kutta-Nyström methods for the hyperbolic equation. We prove the order optimality of a class of block preconditioners for the stage equation system arising from these problems, and furthermore we show that the LD and DU preconditioners of Rana et al. are in this class. We carry out numerical experiments on several test problems in this class-the 2D diffusion equation, Pennes bioheat equation, the wave equation, and the Klein-Gordon equation, with both constant and variable coefficients. Our experiments show that these preconditioners, particularly the LD preconditioner, are successful at reducing the condition number of the systems as well as improving the convergence rate and solve time for GMRES applied to the stage equations.
Numerical Methods for Partial Differential Equations, 2011
The PDE part of the bidomain equations is discretized in time with fully implicit Runge-Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time stepping operator in the proper Sobolev spaces we show that the preconditioned systems have bounded condition numbers given that the Runge-Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order-optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept weakly positive definite matrices is introduced and analyzed.
A parallelizable preconditioner for the iterative solution of implicit Runge–Kutta-type methods
Journal of Computational and Applied Mathematics, 1999
The main di culty in the implementation of most standard implicit Runge-Kutta (IRK) methods applied to (sti ) ordinary di erential equations (ODEs) is to e ciently solve the nonlinear system of equations. In this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit Euler method. The preconditioner is based on the W-transformation of the RK coe cient matrices discovered by Hairer and Wanner. For sti ODEs the preconditioner is by construction asymptotically exact for methods with an invertible RK coe cient matrix. The methodology is particularly useful when applied to super partitioned additive Runge-Kutta (SPARK) methods. The nonlinear system can be solved by inexact simpliÿed Newton iterations: at each simpliÿed Newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system.
Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations
Siam Journal on Matrix Analysis and Applications, 2011
In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computational cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its convergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution.
Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations
2009
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution.
Preconditioners for the Discontinuous Galerkin Time-Stepping Method of Arbitrary Order
ESAIM: Mathematical Modelling and Numerical Analysis, 2017
We develop a preconditioner for systems arising from space-time finite element discretizations of parabolic equations. The preconditioner is based on a transformation of the coupled system into block diagonal form and an efficient solution strategy for the arising 2 × 2 blocks. The suggested strategy makes use of an inexact factorization of the Schur complement of these blocks, for which uniform bounds on the condition number can be proven. The main computational effort of the preconditioner lies in solving implicit Euler-like problems, which allows for the usage of efficient standard solvers. Numerical experiments are performed to corroborate our theoretical findings.