Implementing Explicit Runge Kutta Methods

.gitignore

@@ -7,3 +7,5 @@ docs/build/
 Manifest.toml
 tmp/
 .vscode/
+*.vtu
+*.pvd

my_tests/TransientFEsTests/BoundaryHeatEquationTests.jl

@@ -0,0 +1,97 @@
+using Pkg
+
+
+Pkg.add(url="https://github.com/tamaratambyah/Gridap.jl", rev="rungekutta")
+
+using Gridap
+
+using Test
+
+# Analytical functions
+# u(x,t) = (x[1]+x[2])*t
+# u(x,t) = (2*x[1]+x[2])*t
+u(x,t) = (1.0-x[1])*x[1]*(1.0-x[2])*x[2]*t
+# u(x,t) = (1.0-x[1])*x[1]*t
+
+u(t) = x -> u(x,t)
+∂tu = ∂t(u)
+f(t) = x -> ∂t(u)(x,t)-Δ(u(t))(x)
+
+
+
+domain = (0,1,0,1)
+partition = (2,2)
+model = CartesianDiscreteModel(domain,partition)
+
+order = 2
+
+reffe = ReferenceFE(lagrangian,Float64,order)
+V0 = FESpace(
+  model,
+  reffe,
+  conformity=:H1,
+  dirichlet_tags=[1,2,3,4,5,6]
+)
+U = TransientTrialFESpace(V0,u)
+
+Ω = Triangulation(model)
+degree = 2*order
+dΩ = Measure(Ω,degree)
+
+neumanntags = [7,8]
+Γ = BoundaryTriangulation(model,tags=neumanntags)
+dΓ = Measure(Γ,degree)
+nb = get_normal_vector(Γ)
+
+#
+a(u,v) = ∫(∇(v)⋅∇(u))dΩ
+b(v,t) = ∫(v*f(t))dΩ
+b_Γ(v,t) = ∫(v*(∇(u(t))⋅nb))dΓ
+
+lhs(t,u,v) = ∫( v* (u) )dΩ
+rhs(t,u,v) = b(v,t) + b_Γ(v,t) - a(u,v)
+
+jac(t,u,du,v) = ∫(( ∇(v)⊙∇(du) ))dΩ
+jac_t(t,u,dut,v) = ∫( dut*v )dΩ
+
+op = TransientEXRungeKuttaFEOperator(lhs,rhs,jac,jac_t,U,V0)
+
+t0 = 0.0
+tF = 10.0
+dt = 0.001
+
+U0 = U(0.0)
+uh0 = interpolate_everywhere(u(0.0),U0)
+
+ls = LUSolver()
+
+# ode_solver = EXRungeKutta(ls,dt,:EX_FE_1_0_1)
+ode_solver = EXRungeKutta(ls,dt,:EX_SSP_3_0_3)
+
+sol_t = solve(ode_solver,op,uh0,t0,tF)
+
+l2(w) = w*w
+
+tol = 1.0e-6
+
+errors_rk_fe = []
+ts_rk_fe = []
+for (uh_tn, tn) in sol_t
+  e = u(tn) - uh_tn
+  el2 = sqrt(sum( ∫(l2(e))dΩ )) / ( sqrt( sum( ∫( u(tn) )dΩ  )) )
+  println(el2)
+  (@test el2 < tol)
+  errors_rk_fe = [errors_rk_fe; el2]
+  ts_rk_fe = [ts_rk_fe; tn]
+end
+
+plot(ts_rk_fe,errors_rk_fe)
+plot!(
+  #shape=:auto,
+  xlabel="t",
+  ylabel="relative error",
+  title="RK-FE"
+  # yaxis=:log
+  )
+plot!(show=true)
+savefig(string("rk_fe_error_boundary"))

my_tests/TransientFEsTests/DGHeatEquationTests.jl

@@ -0,0 +1,100 @@
+using Pkg
+using Test
+Pkg.add(url="https://github.com/tamaratambyah/Gridap.jl", rev="rungekutta")
+
+using Gridap
+
+
+
+u(x,t) = (1.0-x[1])*x[1]*(1.0-x[2])*x[2]*t
+u(t::Real) = x -> u(x,t)
+f(t) = x -> ∂t(u)(x,t)-Δ(u(t))(x)
+
+L= 1.0
+n = 2
+domain = (0,L,0,L)
+partition = (n,n)
+model = CartesianDiscreteModel(domain,partition)
+
+order = 2
+
+reffe = ReferenceFE(lagrangian,Float64,order)
+V0 = FESpace(
+  model,
+  reffe,
+  conformity=:L2
+)
+U = TransientTrialFESpace(V0)
+
+Ω = Triangulation(model)
+degree = 2*order
+dΩ = Measure(Ω,degree)
+
+Γ = BoundaryTriangulation(model)
+dΓ = Measure(Γ,degree)
+nb = get_normal_vector(Γ)
+
+Λ = SkeletonTriangulation(model)
+dΛ = Measure(Λ,degree)
+ns = get_normal_vector(Λ)
+
+a(u,v) = ∫(∇(v)⋅∇(u))dΩ
+b(v,t) = ∫(v*f(t))dΩ
+m(u,v) = ∫(v*u)dΩ
+
+h = 1.0 / n
+γ = order*(order+1)
+a_Γ(u,v) = ∫( (γ/h)*v*u - v*(∇(u)⋅nb) - (∇(v)⋅nb)*u )dΓ
+b_Γ(v,t) = ∫( (γ/h)*v*u(t) - (∇(v)⋅nb)*u(t) )dΓ
+
+a_Λ(u,v) = ∫( (γ/h)*jump(v*ns)⊙jump(u*ns) - jump(v*ns)⊙mean(∇(u)) - mean(∇(v))⊙jump(u*ns) )dΛ
+
+lhs(t,u,v) = m(u,v)
+rhs(t,u,v) = b(v,t) + b_Γ(v,t) - a(u,v) - a_Γ(u,v) - a_Λ(u,v)
+jac(t,u,du,v) = a(du,v) + a_Γ(du,v) + a_Λ(du,v)
+jac_t(t,u,dut,v) = m(dut,v)
+
+op = TransientEXRungeKuttaFEOperator(lhs,rhs,jac,jac_t,U,V0)
+
+t0 = 0.0
+tF = 10.0
+dt = 0.001
+
+U0 = U(0.0)
+uh0 = interpolate_everywhere(u(0.0),U0)
+
+ls = LUSolver()
+# ode_solver = EXRungeKutta(ls,dt,:EX_FE_1_0_1)
+ode_solver = EXRungeKutta(ls,dt,:EX_SSP_3_0_3)
+
+sol_t = solve(ode_solver,op,uh0,t0,tF)
+
+l2(w) = w*w
+
+tol = 1.0e-6
+
+errors_rk_fe = []
+ts_rk_fe = []
+for (uh_tn, tn) in sol_t
+
+  e = u(tn) - uh_tn
+  el2 = sqrt(sum( ∫(l2(e))dΩ )) #/ ( sqrt(sum( ∫(l2( u(tn) ))dΩ ))  )
+  println(el2, tn)
+  @test el2 < tol
+
+  errors_rk_fe = [errors_rk_fe; el2]
+  ts_rk_fe = [ts_rk_fe; tn]
+end
+
+
+
+plot(ts_rk_fe,errors_rk_fe)
+plot!(
+  #shape=:auto,
+  xlabel="t",
+  ylabel="relative error",
+  title="RK-FE"
+  # yaxis=:log
+  )
+plot!(show=true)
+savefig(string("rk_fe_error_dG"))

my_tests/TransientFEsTests/HeatEquationTest_1D.jl

@@ -0,0 +1,91 @@
+using Pkg
+Pkg.add(url="https://github.com/tamaratambyah/Gridap.jl", rev="rungekutta")
+
+using Gridap
+using Plots
+using Test
+
+u(x,t) = x[1]*(1-x[1])*t
+u(t) = x -> u(x,t)
+∂tu = ∂t(u)
+f(t) = x -> ∂t(u)(x,t)-Δ(u(t))(x)
+
+
+domain = (0,1)
+partition = (2)
+model = CartesianDiscreteModel(domain,partition)
+
+order = 2
+
+reffe = ReferenceFE(lagrangian,Float64,order)
+V0 = FESpace(
+  model,
+  reffe,
+  conformity=:H1,
+  dirichlet_tags="boundary"
+)
+U = TransientTrialFESpace(V0,u)
+
+Ω = Triangulation(model)
+degree = 2*order
+dΩ = Measure(Ω,degree)
+
+#
+lhs(t,u,v) = ∫( v* (u) )dΩ
+rhs(t,u,v) = ∫(v*f(t))dΩ -  ∫(( ∇(v)⊙∇(u) ))dΩ
+jac(t,u,du,v) = ∫(( ∇(v)⊙∇(du) ))dΩ
+jac_t(t,u,dut,v) = ∫( dut*v )dΩ
+
+op = TransientEXRungeKuttaFEOperator(lhs,rhs,jac,jac_t,U,V0)
+
+t0 = 0.0
+tF = 10.0
+dt = 0.001
+
+uh0 = interpolate_everywhere(u(0.0),U(0.0))
+
+ls = LUSolver()
+# ode_solver = EXRungeKutta(ls,dt,:EX_FE_1_0_1)
+# ode_solver = EXRungeKutta(ls,dt,:EX_SSP_3_0_3)
+ode_solver = DIRungeKutta(ls,dt,:BE_1_0_1)
+sol_t = solve(ode_solver,op,uh0,t0,tF)
+
+
+l2(w) = w*w
+
+tol = 1.0e-6
+
+errors_rk_fe = []
+ts_rk_fe = []
+for (uh_tn, tn) in sol_t
+  e = interpolate_everywhere(u(tn),U(tn)) - uh_tn
+  el2 = sqrt(sum( ∫(l2(e))dΩ ))
+  println(el2)
+  @test el2< tol
+  @test ~any( isnan.(get_free_dof_values(uh_tn)))
+
+  errors_rk_fe = [errors_rk_fe; el2]
+  ts_rk_fe = [ts_rk_fe; tn]
+end
+
+
+
+plot(ts_rk_fe,errors_rk_fe)
+plot!(
+  #shape=:auto,
+  xlabel="t",
+  ylabel="relative error",
+  title="RK-FE"
+  # yaxis=:log
+  )
+plot!(show=true)
+savefig(string("rk_fe_error"))
+
+
+
+sol_t = solve(ode_solver,op,uh0,t0,tF)
+createpvd("my_tests/transient_sol/poisson_transient_solution") do pvd
+  for (uh,t) in sol_t
+    pvd[t] = createvtk(Ω,"my_tests/transient_sol/poisson_transient_solution_$t"*".vtu",cellfields=["u"=>uh])
+  end
+end

my_tests/TransientFEsTests/HeatEquationTest_2D.jl

@@ -0,0 +1,87 @@
+using Pkg
+Pkg.add(url="https://github.com/tamaratambyah/Gridap.jl", rev="rungekutta")
+
+using Gridap
+using Plots
+using Test
+
+u(x,t) = (1.0-x[1])*x[1]*(1.0-x[2])*x[2]*t
+u(t) = x -> u(x,t)
+∂tu = ∂t(u)
+f(t) = x -> ∂t(u)(x,t)-Δ(u(t))(x)
+
+
+domain = (0,1,0,1)
+partition = (2,2)
+model = CartesianDiscreteModel(domain,partition)
+
+order = 2
+
+reffe = ReferenceFE(lagrangian,Float64,order)
+V0 = FESpace(
+  model,
+  reffe,
+  conformity=:H1,
+  dirichlet_tags="boundary"
+)
+U = TransientTrialFESpace(V0,u)
+
+Ω = Triangulation(model)
+degree = 2*order
+dΩ = Measure(Ω,degree)
+
+#
+lhs(t,u,v) = ∫( v* (u) )dΩ
+rhs(t,u,v) = ∫(v*f(t))dΩ -  ∫(( ∇(v)⊙∇(u) ))dΩ
+jac(t,u,du,v) = ∫(( ∇(v)⊙∇(du) ))dΩ
+jac_t(t,u,dut,v) = ∫( dut*v )dΩ
+
+op = TransientEXRungeKuttaFEOperator(lhs,rhs,jac,jac_t,U,V0)
+
+t0 = 0.0
+tF = 1.0
+dt = 0.001
+
+uh0 = interpolate_everywhere(u(0.0),U(0.0))
+
+ls = LUSolver()
+# ode_solver = EXRungeKutta(ls,dt,:EX_FE_1_0_1)
+ode_solver = EXRungeKutta(ls,dt,:EX_SSP_3_0_3)
+
+sol_t = solve(ode_solver,op,uh0,t0,tF)
+
+
+l2(w) = w*w
+
+tol = 1.0e-6
+
+errors_rk_fe = []
+ts_rk_fe = []
+for (uh_tn, tn) in sol_t
+  e = u(tn) - uh_tn
+  el2 = sqrt(sum( ∫(l2(e))dΩ ))
+  println(el2)
+  @test el2< tol
+
+  errors_rk_fe = [errors_rk_fe; el2]
+  ts_rk_fe = [ts_rk_fe; tn]
+end
+
+
+plot(ts_rk_fe,errors_rk_fe)
+plot!(
+  #shape=:auto,
+  xlabel="t",
+  ylabel="relative error",
+  title="RK-FE"
+  # yaxis=:log
+  )
+plot!(show=true)
+savefig(string("rk_fe_error_2D"))
+
+
+createpvd("my_tests/transient_sol/poisson_transient_solution") do pvd
+  for (uh,t) in sol_t
+    pvd[t] = createvtk(Ω,"my_tests/transient_sol/poisson_transient_solution_$t"*".vtu",cellfields=["u"=>uh])
+  end
+end