Biorthogonal Rosenbrock-Krylov time discretization methods

Abstract

Many scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.

Linearly Implicit Multistep Methods for Time Integration

“…As solving large nonlinear systems can be very expensive, many types of linearly implicit methods have been developed that only require solutions of linear systems at each step. Rosenbrock methods [39] (and their many extensions [20,47,50]) are linearized implicit Runge-Kutta methods. Implicit-explicit (IMEX) methods [1,2,14,38,[54][55][56] couple an implicit scheme for the stiff component with an explicit scheme for the non-stiff component of a split problem.…”

Linearly implicit GARK schemes

“…In contrast to classical interpolation/extrapolation-based multirate Rosenbrock methods [11], generalized multirate Rosenbrock-Wanner schemes have been introduced in [6] as a special instance of partitioned Rosenbrock-W schemes. Matrix-free Rosenbrock-W methods were proposed in [22,33], and Rosenbrock-Krylov methods that approximate the Jacobian in an Arnoldi space in [9,17,[28][29][30][31]. Application of Rosenbrock methods to parabolic partial differential equations, and the avoidance of order reduction, have been discussed in [3,8,16,23].…”