Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs

by   Xiangmin Jiao, et al.

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 semi-implicit RK (aka 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 cost as singly DIRK (SDIRK). We approximately factorize the diagonal blocks in these preconditioners using a near-linear-complexity multilevel ILU factorization called HILUCSI, which is significantly more robust and more efficient than ILU(0). We apply the 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.


page 1

page 2

page 3

page 4


High order semi-implicit multistep methods for time dependent partial differential equations

We consider the construction of semi-implicit linear multistep methods w...

Stage-parallel fully implicit Runge-Kutta implementations with optimal multilevel preconditioners at the scaling limit

We present an implementation of a fully stage-parallel preconditioner fo...

Symmetric Near-Field Schur Complement Preconditioner for Hierarchal Electric Field Integral Equation Solve

In this paper, a robust and effective preconditioner for the fast Method...

A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE

A new preconditioner based on a block LDU factorization with algebraic m...

Nearly optimal scaling in the SR decomposition

In this paper we analyze the nearly optimal block diagonal scalings of t...

High-order implicit time integration scheme based on Padé expansions

A single-step high-order implicit time integration scheme for the soluti...

Fast stable finite difference schemes for nonlinear cross-diffusion

The dynamics of cross-diffusion models leads to a high computational com...

Please sign up or login with your details

Forgot password? Click here to reset