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Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems. (English) Zbl 1452.65198

Summary: In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge-Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type. We present SRK methods composed of \(L\) ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree \(L\). Internal stability is maintained at large stage number through an ordering algorithm which limits internal amplification factors to \(10 L^2\). Test results for mildly stiff nonlinear advection-diffusion-reaction problems with moderate \((\lesssim1)\) mesh Péclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65L04 Numerical methods for stiff equations

Software:

na20; RKC; gmp; MPFR; GSL; RODAS
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References:

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