Time-stepping

For convenience, we include utilities for two explicit Runge-Kutta schemes. For more advanced time-stepping functionality, we recommend DifferentialEquations.jl.

Carpenter and Kennedy's (4,5) method

ck45() returns coefficients for a low-storage Runge-Kutta method.

Example usage:

using Plots
using StartUpDG

# Brusselator
B = 3
f(y1,y2) = [1+y1^2*y2-(B+1)*y1, B*y1-y1^2*y2]
f(Q) = f(Q[1],Q[2])
p,q = 1.5, 3.0
Q = [p;q]

dt = .1
FinalTime = 20

res = zero.(Q) # init RK residual
rk4a,rk4b,rk4c = ck45()
Nsteps = ceil(FinalTime/dt)
dt = FinalTime/Nsteps

plot() # init plot
for i = 1:Nsteps
    global res # yes, I know...this is just for simplicty
    for INTRK=1:5
        rhsQ = f(Q)
        @. res = rk4a[INTRK]*res + dt*rhsQ # i = RK stage
        @. Q =  Q + rk4b[INTRK]*res
    end
    scatter!([i*dt;i*dt],Q)
end
display(plot!(leg=false))