Spectral methods in time for parabolic problems
SIAM Journal on Numerical Analysis
Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
The pseudospectral method for solving differential eigenvalue problems
Journal of Computational Physics
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Expokit: a software package for computing matrix exponentials
ACM Transactions on Mathematical Software (TOMS)
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
A new class of time discretization schemes for the solution of nonlinear PDEs
Journal of Computational Physics
Spectral methods in MatLab
Exponential time differencing for stiff systems
Journal of Computational Physics
A Schur-Parlett Algorithm for Computing Matrix Functions
SIAM Journal on Matrix Analysis and Applications
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
Journal of Computational Physics
Comparison of methods for evaluating functions of a matrix exponential
Applied Numerical Mathematics
Journal of Computational Physics
Hi-index | 31.46 |
Over the last decade there has been renewed interest in applying exponential time differencing (ETD) time stepping schemes to the solution of stiff systems. In this paper, we present an implementation of such a scheme to the fully spectral solution of the incompressible magnetohydrodynamic equations in a spherical shell. One problem associated with ETD schemes is the accurate calculation of the necessary matrices; we implement and discuss in detail a variety of different methods including direct computation, contour integration, spectral expansions and recurrence relations. We compare the accuracy of six different second-order methods in determining the evolution of a three-dimensional magnetic field under the action of a prescribed time-dependent flow of electrically conducting fluid, and find that for the timestep restriction imposed by the nonlinear terms, ETD methods are no more accurate than linearly implicit methods which have the significant advantage of being easier to implement. However, ETD methods are more readily extendable than those which are linearly implicit and will become much more advantageous at higher order.