Operator splitting implicit integration factor methods for stiff reaction–diffusion–advection systems

Abstract

For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction-diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high-order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction-diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.

Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations

“…We consider a fourth-order system of equations with stiff reactions on a two-dimensional domain Ω = (0, 2π) 2 . Similar examples of second-order systems of reaction-diffusion and ADR equations were constructed in [20,31,32] and used to test the IIF and Krylov IIF schemes for solving problems with stiff reactions. The fourth-order system has the following form:…”

“…For example, nonlinear advection can dominate at early times in the system, and later the diffusion may become dominant [30]. In [31], operator splitting compact IIF methods were designed to solve stiff ADR systems. The multistep and single-step Krylov IIF methods were developed to solve stiff ADR systems in [32,33].…”

Numerical simulations of a reduced model for blood coagulation

“…To deal with all these potential causes of numerical instabilities, we use a second-order operator (Strang) splitting method [25,27,30,50,55] and split (2) into two simpler subproblems-the reaction and the advection-diffusion (AD) one [8]-that are solved sequentially on each time interval [t n , t n+1 ].…”

Semi-implicit integration factor methods on sparse grids for high-dimensional systems

“…This property results in good efficiency, in addition to excellent stability conditions (e.g., the second-order IIF is linearly unconditionally stable). Moreover, the IIF method can handle reaction-convection-diffusion equations through an operator splitting technique [26] and can be incorporated with the adaptive meshes and general curvilinear coordinates [27]. Because the exact treatment of the diffusion terms requires computing exponentials of matrices resulting from the discretization of the linear differential operators, a compact representation of IIF (cIIF) [28, 27] and an array representation of IIF (AcIIF) [29] for systems with high spatial dimensions have been introduced.…”