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Fast high-order compact exponential time differencing Runge-Kutta methods for second-order semilinear parabolic equations. (English) Zbl 1342.65187

Summary: In this paper we propose fast high-order numerical methods for solving a class of second-order semilinear parabolic equations in regular domains. The proposed methods are explicit in nature, and use exponential time differencing and Runge-Kutta approximations in combination with a linear splitting technique to achieve accurate and stable time integration. A two-step compact difference scheme is employed for spatial discretization to obtain fourth-order accuracy and make use of fast Fourier transform-based fast calculations. Such methods can be applied to problems with stiff nonlinearities and boundary conditions of Dirichlet or periodic types. Linear stability analysis and various numerical experiments are also presented to demonstrate accuracy and stability of the proposed methods.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K58 Semilinear parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65T50 Numerical methods for discrete and fast Fourier transforms

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