Fourth-Order Time-Stepping for Stiff PDEs

Fourth-Order Time-Stepping for Stiff PDEs

A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competi...

Explicit Time-Stepping for Stiff ODEs

We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvemen...

High order schemes based on operator splitting and deferred corrections for stiff time dependent PDEs

We consider quadrature formulas of high order in time based on Radau–type, L–stable implicit Runge–Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs iterative deferred corrections to compute the solution at the collocation nodes of the quadrature formulas. The numerical stability is guaranteed by a dedic...

Fourth-order parallel rosenbrock formulae for stiff systems

K e y w o r d s R o s e n b r o c k methods, A-stable, Parallel algorithm, Stiff initial value problem, Adal> tivity. 1. I N T R O D U C T I O N We consider the numerical solution of systems of initial value ordinary differential equations (ODEs), i.e., initial value problems (IVPs), of the form y'(t) -f (y(t)) , y(to) = Yo, (1) where y : R --* R "~ and f : R "~ --* R m. Runge-Kutta methods app...

Traveling Wave Solutions of Fourth Order PDEs for Image Processing