Abstract
Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.
Highlights
Efficient and high-order accuracy temporal numerical methods are important for accurate numerical simulations of time-dependent problems
Integration factor (IF) methods are a class of “exactly linear part” time-discretization methods. They are designed for solving nonlinear partial differential equations (PDEs) whose highest spatial derivatives are linear
We develop Krylov integration factor (IIF) methods for solving semilinear fourth-order PDEs
Summary
Efficient and high-order accuracy temporal numerical methods are important for accurate numerical simulations of time-dependent problems. In [20], a class of efficient implicit integration factor (IIF) methods were developed to solve reaction–diffusion systems that have both stiff linear diffusion and stiff nonlinear reaction terms. We develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations These are solved by implicit schemes in order to obtain large time-step sizes. Depending on the accuracy order of IIF time discretization, the second- or fourth-order central finite difference scheme is used to discretize the diffusion terms.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
