Roadmap KdV eq.

[cwittens]

The following I want to do next regarding the KdV eq.

• Add KdV to Benchmark results, see added two kdv_1d to benchmarking tests #208
• add nondim2prim + prim2nondim functions to be able to compare results from $$u_t + u u_x + u_{xxx}=0$$ PDE to the current implementation $$\eta_t+\sqrt{g D} \eta_x+3 / 2 \sqrt{g / D} \eta \eta_x+1 / 6 \sqrt{g D} D^2 \eta_{x x x} = 0$$, see Added kdv_1d non dimensional solitary wave example #217
• add the linear dispersion relation $\omega(k) = \sqrt{g h_0}k(1 + \frac 3 2 \frac{\eta_0}{h_0}) - \frac 1 6 h_0^2k^3$ to https://github.com/NumericalMathematics/DispersiveShallowWater.jl/blob/a5e6adb4319151d9b20139005bc3f95334f8e7b7/src/dispersion_relation.jl, see Add dispersion relation for KdVEquation1D #230.
• add more conversion_functions, right now only waterheight is working, see Added energy_total to KdV and more unit tests #231
• possibly add more examples, see Added energy_total to KdV and more unit tests #231
• more unit tests, see Added energy_total to KdV and more unit tests #231
• more docs
• split up the ode - make it possible to use IMEX in DSW.jl, see also Allow IMEX solvers for stiff problems #165

[JoshuaLampert]

Regarding the conversion functions: I think this basically boils down to implementing energy_total, which should return $\frac 1 2\eta^2$ (or probably $\frac 1 2 g\eta^2$?)and entropy, which computes the same (just forwarding to energy_total similarly to AbstractShallowWaterEquations). This is what we would like to see conserved by the semidiscretization, i.e. checking this by adding entropy to the AnalysisCallbacks of some examples would be good and maybe adding another example using relaxation for fully-discrete entropy-conservation. Finally, we should extend the unit tests for the KdV equation similarly to the BBM equation

DispersiveShallowWater.jl/test/test_unit.jl

Lines 131 to 177 in a5e6adb

	@testitem "BBMEquation1D" setup=[Setup] begin
	equations = @test_nowarn @inferred BBMEquation1D(gravity = 1.0)
	@test_nowarn print(equations)
	@test_nowarn display(equations)
	conversion_functions = [
	waterheight_total,
	waterheight,
	entropy,
	energy_total,
	prim2cons,
	prim2prim,
	prim2phys,
	energy_total_modified,
	entropy_modified,
	hamiltonian
	]
	for conversion in conversion_functions
	@test DispersiveShallowWater.varnames(conversion, equations) isa Tuple
	end
	q = [42.0]
	@test @inferred(prim2prim(q, equations)) == q
	@test isapprox(@inferred(cons2prim(prim2cons(q, equations), equations)), q)
	@test @inferred(waterheight_total(q, equations)) == 42.0
	@test @inferred(waterheight(q, equations)) == 43.0
	@test @inferred(still_water_surface(q, equations)) == 0.0
	@test @inferred(prim2phys(q, equations)) == @inferred(prim2prim(q, equations))
	@testset "energy_total_modified and hamiltonian" begin
	initial_condition = initial_condition_manufactured
	boundary_conditions = boundary_condition_periodic
	mesh = @inferred Mesh1D(-1.0, 1.0, 10)
	solver = Solver(mesh, 4)
	semi = @inferred Semidiscretization(mesh, equations, initial_condition,
	solver; boundary_conditions)
	q = @inferred DispersiveShallowWater.compute_coefficients(initial_condition,
	0.0, semi)
	_, _, _, cache = @inferred DispersiveShallowWater.mesh_equations_solver_cache(semi)
	e_modified = @inferred energy_total_modified(q, equations, cache)
	e_modified_total = @inferred DispersiveShallowWater.integrate(e_modified, semi)
	@test isapprox(e_modified_total, 3.710663574870101)
	U_modified = @inferred entropy_modified(q, equations, cache)
	U_modified_total = @inferred DispersiveShallowWater.integrate(U_modified, semi)
	@test isapprox(U_modified_total, e_modified_total)
	h = @inferred hamiltonian(q, equations, cache)
	h_total = @inferred DispersiveShallowWater.integrate(h, semi)
	@test isapprox(h_total, 0.5)
	end
	end

Functions like prim2prim, prim2cons, and prim2phys should already work out of the box. The others (energy_total_modified, entropy_modified, and hamiltonian) should not be needed for the KdV equation.