Data for: Runge-Kutta-Gegenbauer methods for advection-diffusion problems

frkg2.tgz is a C implementation of FRKG numerical integration at second order. The code consists of two files: the file frkgcore.c contains the code necessary to obtain a solution to the problem prescribed in frkguser.c. The files Tables_N1-N2/arks___.csv contain the coefficents for FRKG schemes at orders N=1...2 and, at each order, over a range of 18 Gegenbauer coefficient values running from 0 to 2N. The coefficients for each scheme is provided for up to Nx257 stages.

The default problem is similar to the Brusselator presented by Hairer & Wanner (see (1.6) of Sect. IV.1): a stiff nonlinear diffusion-reaction problem describing chemical kinetics of a tri-molecular chemical reaction subject to periodic boundary conditions. Advection is added following Abdulle & Vilmart (2013). The problem is specified via three user defined functions: deriv() - populates f(W)=W_t; spec() - returns spectral radius; and Wexact(). User parameters are: RTOL - error tolerence; NE - number of grid points (including boundary) assuming a uniform square mesh; TEND; XMAX; XMIN. To apply the FRKG scheme to other problems it should only be necessary to supply a modified version of frkguser.c.

Note that complex variables are used via complex.h internally to the time-marching function trystep(). Above trystep() no complex variables are visible.

Usage: to compile executable frkg using gcc
gcc -O3 frkgcore.c frkguser.c -lm -o frkg

Solution is written to sol.dat
Reference solution is sol.ref
To plot first species via gnuplot: splot "sol.dat" every 10:10 u 1:2:3 w pm3d

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The additional files Tables_N3-N8/arks___.csv contain the coefficents for FRKG schemes at orders N=3...8 and, at each order, over a range of 18 Gegenbauer coefficient values running from 0 to 2N. The coefficients for each scheme is provided for up to Nx257 stages.