Compact integration factor methods in high spatial dimensions

Abstract

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 N × N spatial points, the exponential matrix is of a size of N 2 × N 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.

Semi-implicit spectral collocation methods for reaction-diffusion equations on annuli

“…The diagonal matrix R has the same form (2.6), but the nodes r i are uniformly distributed. Similar finite difference schemes for the Cartesian geometry can be found in [3,6]. Finite difference approximations based on nonuniform grids are possible but significantly more complicated.…”

“…Similar (second and even higher order) numerical schemes based on finite difference spatial discretizations for reaction-diffusion systems in Cartesian geometry are derived in [3,6]. However, we shall consider mainly the following second order methods (in polar geometry) based on integrating factors.…”

“…Exponential time differencing methods [4,5], integrating factor methods [3,6], and operator splitting methods [7] have all been extensively investigated. Combined with a posterior error estimates, reaction-diffusion systems can be solved adaptively through operator splitting methods [8].…”

“…In [14], an integration factor splitting method was investigated and used for the shallow water equations. Integrating factor finite difference methods were developed in [3,6] for solving reaction-diffusion equations in Cartesian geometries. In this paper, we combine integrating factor methods and spectral collocation methods (Chebyshev in the radial direction and Fourier in the azimuthal direction) to develop numerical schemes for reaction-diffusion problems on annuli.…”

Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations

“…To overcome the challenge of implementing IF-type methods for high-spatial-dimension problems, compact IIF methods [21] were developed for spatial discretizations on rectangular meshes. In [22], compact IIF methods were designed to solve problems in curvilinear coordinates, such as polar and spherical coordinates.…”