Added Algorithm FineRKN4

[HenryLangner]

Added the 4th order Nyström-Method presented by J. M. Fine in the article "Low Order Practical Runge-Kutta-Nyström Methods" (Also linked here: #677). Works with adaptive step-size. The Method passes the implemented convergence tests (harm. Oscillator):

using OrdinaryDiffEq, Test, RecursiveArrayTools, DiffEqDevTools, Statistics

u0 = fill(0.0, 2);
v0 = ones(2);
function f1_harmonic(dv, v, u, p, t)
    dv .= -u
end;
function f2_harmonic(du, v, u, p, t)
    du .= v
end;
function harmonic_analytic(y0, p, x)
    v0, u0 = y0.x
    ArrayPartition(-u0 * sin(x) + v0 * cos(x), u0 * cos(x) + v0 * sin(x))
end;
ff_harmonic = DynamicalODEFunction(f1_harmonic, f2_harmonic; analytic = harmonic_analytic);
prob = DynamicalODEProblem(ff_harmonic, v0, u0, (0.0, 5.0));

dts = 1.0 ./ 2.0 .^ (5:-1:0);

println("##### In Place #####")
sim = test_convergence(dts, prob, FineRKN4(), dense_errors = true);
println(@test sim.𝒪est[:l2]≈5 rtol=1e-1)
println(@test sim.𝒪est[:L2]≈4 rtol=1e-1)
println("with:")
println(sim.𝒪est)
println("##### Out of Place ##### ")

u0 = 0.0;
v0 = 1.0;
function f1_harmonic_nip(v, u, p, t)
    -u
end;
function f2_harmonic_nip(v, u, p, t)
    v
end;

ff_harmonic_nip = DynamicalODEFunction(f1_harmonic_nip, f2_harmonic_nip;
                                       analytic = harmonic_analytic);
prob = DynamicalODEProblem(ff_harmonic_nip, v0, u0, (0.0, 5.0));

csim = test_convergence(dts, prob, FineRKN4(), dense_errors = true);
println(@test sim.𝒪est[:l2]≈5 rtol=1e-1)
println(@test sim.𝒪est[:L2]≈4 rtol=1e-1)
println("with:")
println(sim.𝒪est)

Yields:

There is one more PR on the way which adds adaptive step-size to FineRKN5.
Once these two PRs are merged I will add convergence test for FineRKN4, FineRKN5 and Nystrom4 regarding the damped_oscillator introduced in #1973.

CC @ranocha

[ranocha]

Thanks! I just have a minor comment

[ranocha]

Do you have some tests with adaptive step size control here?

[HenryLangner]

Do you have some tests with adaptive step size control here?

I just added them!

[ranocha]

Thanks!