Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems

Abstract 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 (≲1) mesh Peclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.

[1]  A. Medovikov.  High order explicit methods for parabolic equations , 1998 .

[2]  Willem Hundsdorfer,et al.  RKC time-stepping for advection-diffusion-reaction problems , 2004 .

[3]  Assyr Abdulle,et al.  Second order Chebyshev methods based on orthogonal polynomials , 2001, Numerische Mathematik.

[4]  L. Shampine,et al.  RKC: an explicit solver for parabolic PDEs , 1998 .

[5]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[6]  Dinshaw S. Balsara,et al.  A stabilized Runge-Kutta-Legendre method for explicit super-time-stepping of parabolic and mixed equations , 2014, J. Comput. Phys..

[7]  Matematik,et al.  Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

[8]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[9]  A. Ostermann,et al.  High order splitting methods for analytic semigroups exist , 2009 .

[10]  Assyr Abdulle,et al.  PIROCK: A swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise , 2013, J. Comput. Phys..

[11]  Bram van Leer,et al.  Upwind-difference methods for aerodynamic problems governed by the Euler equations , 1985 .

[12]  Kjell Gustafsson,et al.  Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods , 1991, TOMS.

[13]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[14]  Stephen O'Sullivan,et al.  A class of high-order Runge-Kutta-Chebyshev stability polynomials , 2015, J. Comput. Phys..

[15]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[16]  T. Aslam,et al.  A second-order accurate Super TimeStepping formulation for anisotropic thermal conduction , 2012 .

[17]  Alexander Ostermann,et al.  Interior estimates for time discretizations of parabolic equations , 1995 .

[18]  Stéphane Descombes,et al.  Splitting methods with complex times for parabolic equations , 2009 .

[19]  V. I. Lebedev,et al.  Explicit difference schemes for solving stiff problems with a complex or separable spectrum , 2000 .

[20]  S. O'Sullivan.  Factorized Runge–Kutta–Chebyshev Methods , 2017, 1702.03818.

[21]  Vincent Lefèvre,et al.  MPFR: A multiple-precision binary floating-point library with correct rounding , 2007, TOMS.

[22]  Fernando Casas,et al.  Optimized high-order splitting methods for some classes of parabolic equations , 2011, Math. Comput..

[23]  S. SIAMJ.,et al.  FOURTH ORDER CHEBYSHEV METHODS WITH RECURRENCE RELATION∗ , 2002 .

[24]  G. Nicolis,et al.  Chemical instabilities and sustained oscillations. , 1971, Journal of theoretical biology.

[25]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[26]  Giuseppe Fiorentino,et al.  Design, analysis, and implementation of a multiprecision polynomial rootfinder , 2000, Numerical Algorithms.

[27]  J. Martín-Vaquero,et al.  Extrapolated stabilized explicit Runge-Kutta methods , 2016, J. Comput. Phys..