Preconditioning of implicit Runge-Kutta methods

Abstract

A major problem in obtaining an efficient implementation of fully implicit
Runge-Kutta (IRK) methods applied to systems of differential equations
is to solve the underlying systems of nonlinear equations. Their solution
is usually obtained by application of modified Newton iterations with
an approximate Jacobian matrix. The systems of linear equations of the
modified Newton method can actually be solved approximately with a
preconditioned linear iterative method. In this article we present
a truly parallelizable preconditioner to the approximate Jacobian matrix.
Its decomposition cost for a sequential or parallel implementation can
be made equivalent to the cost corresponding to the implicit Euler method.
The application of the preconditioner to a vector
consists of three steps: two steps involve the solution of a linear system
with the same block-diagonal matrix and one step involves a matrix-vector
product. The preconditioner is asymptotically correct for the Dahlquist
test equation. Some free parameters of the preconditioner can be determined
in order to optimize certain properties of the preconditioned approximate
Jacobian matrix.