A method for exponential propagation of large systems of stiff nonlinear differential equations
Journal of Scientific Computing
Krylov methods for the incompressible Navier-Stokes equations
Journal of Computational Physics
Removing the stiffness from interfacial flows with surface tension
Journal of Computational Physics
Microstructural evolution in inhomogeneous elastic media
Journal of Computational Physics
On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases
Journal of Computational Physics
A new class of time discretization schemes for the solution of nonlinear PDEs
Journal of Computational Physics
Microstructural evolution in orthotropic elastic media
Journal of Computational Physics
Exponential time differencing for stiff systems
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Efficient semi-implicit schemes for stiff systems
Journal of Computational Physics
Compact integration factor methods for complex domains and adaptive mesh refinement
Journal of Computational Physics
Journal of Computational Physics
A robust and efficient method for steady state patterns in reaction-diffusion systems
Journal of Computational Physics
Array-representation integration factor method for high-dimensional systems
Journal of Computational Physics
Hi-index | 31.47 |
The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of NxN spatial points, the exponential matrix is of a size of N^2xN^2 based on direct representations. The vector-matrix multiplication is of O(N^4), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two- and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two- or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N^2) and O(N^3), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.