Fix type stabilities for FVMSystems

[DanielVandH]

This was slowing things down significantly.

[codecov]

Codecov Report

Patch coverage: 98.44% and project coverage change: +0.76% 🎉

Comparison is base (df43599) 92.98% compared to head (71ffbc3) 93.75%.
Report is 2 commits behind head on main.

❗ Current head 71ffbc3 differs from pull request most recent head 6cf7a45. Consider uploading reports for the commit 6cf7a45 to get more accurate results

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #48      +/-   ##
==========================================
+ Coverage   92.98%   93.75%   +0.76%     
==========================================
  Files          20       20              
  Lines         998     1057      +59     
==========================================
+ Hits          928      991      +63     
+ Misses         70       66       -4     

Files Changed	Coverage Δ
src/problem.jl	94.76% <96.82%> (+4.32%)	⬆️
src/conditions.jl	99.13% <100.00%> (-0.04%)	⬇️
src/equations/boundary_edge_contributions.jl	98.48% <100.00%> (+0.12%)	⬆️
src/equations/individual_flux_contributions.jl	100.00% <100.00%> (ø)
src/utils.jl	83.78% <100.00%> (+15.36%)	⬆️

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[DanielVandH]

Just to show the difference: Consider this benchmark

using FiniteVolumeMethod, OrdinaryDiffEq, LinearSolve, BenchmarkTools, DelaunayTriangulation
const FVM = FiniteVolumeMethod

tri = triangulate_rectangle(0, 1, 0, 1, 25, 25, single_boundary=false)
mesh = FVMGeometry(tri)
Φ_bot = (x, y, t, u, p) -> -1 / 4 * exp(-x - t / 2)
Φ_right = (x, y, t, u, p) -> 1 / 4 * exp(-1 - y - t / 2)
Φ_top = (x, y, t, u, p) -> exp(-1 - x - t / 2)
Φ_left = (x, y, t, u, p) -> -1 / 4 * exp(-y - t / 2)
Φ_bc_fncs = (Φ_bot, Φ_right, Φ_top, Φ_left)
Φ_bc_types = (Neumann, Neumann, Dirichlet, Neumann)
Φ_BCs = BoundaryConditions(mesh, Φ_bc_fncs, Φ_bc_types)
Ψ_bot = (x, y, t, u, p) -> exp(x + t / 2)
Ψ_right = (x, y, t, u, p) -> -1 / 4 * exp(1 + y + t / 2)
Ψ_top = (x, y, t, u, p) -> -1 / 4 * exp(1 + x + t / 2)
Ψ_left = (x, y, t, u, p) -> exp(y + t / 2)
Ψ_bc_fncs = (Ψ_bot, Ψ_right, Ψ_top, Ψ_left)
Ψ_bc_types = (Dirichlet, Neumann, Neumann, Dirichlet)
Ψ_BCs = BoundaryConditions(mesh, Ψ_bc_fncs, Ψ_bc_types)
Φ_q = (x, y, t, α, β, γ, p) -> (-α[1] / 4, -β[1] / 4)
Ψ_q = (x, y, t, α, β, γ, p) -> (-α[2] / 4, -β[2] / 4)
Φ_S = (x, y, t, (Φ, Ψ), p) -> Φ^2 * Ψ - 2Φ
Ψ_S = (x, y, t, (Φ, Ψ), p) -> -Φ^2 * Ψ + Φ
Φ_exact = (x, y, t) -> exp(-x - y - t / 2)
Ψ_exact = (x, y, t) -> exp(x + y + t / 2)
Φ₀ = [Φ_exact(x, y, 0) for (x, y) in each_point(tri)]
Ψ₀ = [Ψ_exact(x, y, 0) for (x, y) in each_point(tri)]
Φ_prob = FVMProblem(mesh, Φ_BCs; flux_function=Φ_q, source_function=Φ_S,
    initial_condition=Φ₀, final_time=5.0)
Ψ_prob = FVMProblem(mesh, Ψ_BCs; flux_function=Ψ_q, source_function=Ψ_S,
    initial_condition=Ψ₀, final_time=5.0)
prob = FVMSystem(Φ_prob, Ψ_prob)
alg = TRBDF2(linsolve=KLUFactorization())
parallel = Val(false)
saveat = 0.1
@benchmark solve($prob, $alg, ; parallel=$parallel, saveat=$saveat)

On this branch:

BenchmarkTools.Trial: 69 samples with 1 evaluation.
 Range (min … max):  68.157 ms … 100.711 ms  ┊ GC (min … max): 0.00% … 28.19%
 Time  (median):     72.281 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   72.922 ms ±   4.346 ms  ┊ GC (mean ± σ):  0.56% ±  3.39%

     ▁▄▄▄▄ ▄▄▄ █▁▄▄█▁█▁  ▁ ▁  █  ▁
  ▆▁▆█████▆███▆████████▁▆█▁█▆▆█▆▆█▁▁▁▁▁▁▁▁▆▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆ ▁
  68.2 ms         Histogram: frequency by time         84.1 ms <

 Memory estimate: 13.16 MiB, allocs estimate: 58802.

Without:

BenchmarkTools.Trial: 2 samples with 1 evaluation.
 Range (min … max):  2.646 s …   2.736 s  ┊ GC (min … max): 19.30% … 20.46%
 Time  (median):     2.691 s              ┊ GC (median):    19.89%
 Time  (mean ± σ):   2.691 s ± 63.664 ms  ┊ GC (mean ± σ):  19.89% ±  0.82%

  █                                                       █  
  █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  2.65 s         Histogram: frequency by time        2.74 s <

 Memory estimate: 6.96 GiB, allocs estimate: 14403237.

Yikes :)