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Fourth-order time-stepping for stiff PDEs. (English) Zbl 1077.65105
Summary: A modification of the exponential time-differencing fourth-order Runge-Kutta method for solving stiff nonlinear partial differential equations (PDEs) is presented that solves the problem of numerical instability in the scheme as proposed by S. M. Cox and P. C. Matthews [J. Comput. Phys. 176, No. 2, 430–455 (2002; Zbl 1005.65069)] and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competing methods of implicit-explicit differencing, integrating factors, time-splitting, and B. Fornberg and T. A. Driscoll’s [ibid. 155, No. 2, 456–467 (1999; Zbl 0937.65109)] “sliders” for the Korteweg-de Vries equation, Kuramoto-Sivashinsky, Burgers, and Allen-Cahn equations in one space dimension. Implementations of the method is illustrated by short MATLAB programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.

MSC:
65M20Method of lines (IVP of PDE)
65L06Multistep, Runge-Kutta, and extrapolation methods
65M70Spectral, collocation and related methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
65M12Stability and convergence of numerical methods (IVP of PDE)
Software:
Matlab