Order optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations (original) (raw)
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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.
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Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs . The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditioners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting. Finally, we improve the condition number by using a generalized Gauss-Seidel preconditioner.
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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.
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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.
An order optimal solver for the discretized bidomain equations
Numerical Linear Algebra with Applications, 2007
The electrical activity in the heart is governed by the Bidomain equations. In this paper we analyze an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss-Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations (PDEs).
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.
Preconditioning Techniques for the Bidomain Equations
In this work we discuss parallel preconditioning techniques for the bidomain equations, a non-linear system of partial differential equations which is widely used for describing electrical activity in cardiac tissue. We focus on the solution of the linear system associated with the elliptic part of the bidomain model, since it dominates computation, with the preconditioned conjugate gradient method. We compare different parallel preconditioning techniques, such as block incomplete LU, additive Schwarz and multigrid. The implementation is based on the PETSc library and we report results for a 16-node HP cluster. The results suggest the multigrid preconditioner is the best option for the bidomain equations.
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.
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2003
We consider the solution by Krylov subspace methods of a certain class of hermitian indefinite linear systems, such as those that arise from discretization of the Stokes equations in incompressible fluid mechanics. We discuss a diagonal preconditioning of these systems that amounts to multiplying some of the equations by while the others are left unchanged. We show that this preconditioning puts all the eigenvalues of the relevant matrix in the open right half plane, enhancing the performance of the Krylov subspace methods in many cases.