% Returns the coefficients used in a RK method as defined by a Butcher Tableau.
%
% @param method - string specifying which Runge-Kutta method to be used.
% @return s - number of stages
% @return a - coefficents for intermediate stages
% @return b - weights for summing stages
% @return c - time step coefficents for intermediate stages
function [s,a,b,c] = butcherTableau(method)
switch method
    case "tvd-3"
        % TVD (Total Variational Diminishing)
        s = 3;
        a = zeros(s,s-1);
        a(2,1) = 1;
        a(3,1) = 1/4; a(3,2) = 1/4;
        b = [1/6, 1/6, 2/3];
        c = [0 1 1/2];
    case "rk4"
        % Standard RK4
        s = 4;
        a = zeros(s,s-1);
        a(2,1) = 1/2;
        a(3,1) = 0; a(3,2) = 1/2;
        a(4,1) = 0; a(4,2) = 0; a(4,3) = 1;
        b = [1/6 1/3 1/3 1/6];
        c = [0, 1/2, 1/2, 1];
    case "rk4-3/8"
        % 3/8 RK4 (Kuttas method). Lower truncation error, more flops.
        % Irreducible, unlike standard rk4.
        s = 4;
        a = zeros(s,s-1);
        a(2,1) = 1/3;
        a(3,1) = -1/3; a(3,2) = 1;
        a(4,1) = 1; a(4,2) = -1; a(4,3) = 1;
        b = [1/8 3/8 3/8 1/8];
        c = [0, 1/3, 2/3, 1];
    case "rk6"
        % Runge-Kutta 6 from Alshina07
        s = 7;
        a = zeros(s,s-1);
        a(2,1) = 4/7;
        a(3,1) = 115/112; a(3,2) = -5/16;
        a(4,1) = 589/630; a(4,2) = 5/18; a(4,3) = -16/45;
        a(5,1) = 229/1200 - 29/6000*sqrt(5); a(5,2) = 119/240 - 187/1200*sqrt(5); a(5,3) = -14/75 + 34/375*sqrt(5); a(5,4) = -3/100*sqrt(5);
        a(6,1) = 71/2400 - 587/12000*sqrt(5); a(6,2) = 187/480 - 391/2400*sqrt(5); a(6,3) = -38/75 + 26/375*sqrt(5); a(6,4) = 27/80 - 3/400*sqrt(5); a(6,5) = (1+sqrt(5))/4;
        a(7,1) = -49/480 + 43/160*sqrt(5); a(7,2) = -425/96 + 51/32*sqrt(5); a(7,3) = 52/15 - 4/5*sqrt(5); a(7,4) = -27/16 + 3/16*sqrt(5); a(7,5) = 5/4 - 3/4*sqrt(5); a(7,6) = 5/2 - 1/2*sqrt(5);
        b = [1/12 0 0 0 5/12 5/12 1/12];
        c = [0, 4/7, 5/7, 6/7, (5-sqrt(5))/10, (5+sqrt(5))/10, 1];
    otherwise
        error('Runge-Kutta method %s is not implemented', method)
end