Merge pull request #306 from tinosulzer/issue-300-Robin-Neumann-bcs

Issue 300 robin neumann bcs

Author: ChrisRackauckas

docs/src/examples.md

@@ -2,8 +2,12 @@
 
 ## Heat equation
 
+### Dirichlet boundary conditions
+
 ```julia
 using ModelingToolkit, DiffEqOperators
+# Method of Manufactured Solutions: exact solution
+u_exact = (x,t) -> exp.(-t) * cos.(x)
 
 # Parameters, variables, and derivatives
 @parameters t x
@@ -13,13 +17,13 @@ using ModelingToolkit, DiffEqOperators
 
 # 1D PDE and boundary conditions
 eq  = Dt(u(t,x)) ~ Dxx(u(t,x))
-bcs = [u(0,x) ~ -x*(x-1)*sin(x),
-        u(t,0) ~ 0.0,
-        u(t,1) ~ 0.0]
+bcs = [u(0,x) ~ cos(x),
+        u(t,0) ~ exp(-t),
+        u(t,1) ~ exp(-t) * cos(1)]
 
 # Space and time domains
 domains = [t ∈ IntervalDomain(0.0,1.0),
-            x ∈ IntervalDomain(0.0,1.0)]
+        x ∈ IntervalDomain(0.0,1.0)]
 
 # PDE system
 pdesys = PDESystem(eq,bcs,domains,[t,x],[u])
@@ -34,5 +38,122 @@ prob = discretize(pdesys,discretization)
 
 # Solve ODE problem
 using OrdinaryDiffEq
-sol = solve(prob,Tsit5(),saveat=0.1)
+sol = solve(prob,Tsit5(),saveat=0.2)
+
+# Plot results and compare with exact solution
+x = prob.space[2]
+t = sol.t
+
+using Plots
+plt = plot()
+for i in 1:length(t)
+    plot!(x,Array(prob.extrapolation[1](t[i])*sol.u[i]),label="Numerical, t=$(t[i])")
+    scatter!(x, u_exact(x, t[i]),label="Exact, t=$(t[i])")
+end
+display(plt)
+```
+### Neumann boundary conditions
+
+```julia
+using ModelingToolkit, DiffEqOperators
+# Method of Manufactured Solutions: exact solution
+u_exact = (x,t) -> exp.(-t) * cos.(x)
+
+# Parameters, variables, and derivatives
+@parameters t x
+@variables u(..)
+@derivatives Dt'~t
+@derivatives Dx'~x
+@derivatives Dxx''~x
+
+# 1D PDE and boundary conditions
+eq  = Dt(u(t,x)) ~ Dxx(u(t,x))
+bcs = [u(0,x) ~ cos(x),
+        Dx(u(t,0)) ~ 0.0,
+        Dx(u(t,1)) ~ -exp(-t) * sin(1)]
+
+# Space and time domains
+domains = [t ∈ IntervalDomain(0.0,1.0),
+        x ∈ IntervalDomain(0.0,1.0)]
+
+# PDE system
+pdesys = PDESystem(eq,bcs,domains,[t,x],[u])
+
+# Method of lines discretization
+# Need a small dx here for accuracy
+dx = 0.01
+order = 2
+discretization = MOLFiniteDifference(dx,order)
+
+# Convert the PDE problem into an ODE problem
+prob = discretize(pdesys,discretization)
+
+# Solve ODE problem
+using OrdinaryDiffEq
+sol = solve(prob,Tsit5(),saveat=0.2)
+
+# Plot results and compare with exact solution
+x = prob.space[2]
+t = sol.t
+
+using Plots
+plt = plot()
+for i in 1:length(t)
+    plot!(x,Array(prob.extrapolation[1](t[i])*sol.u[i]),label="Numerical, t=$(t[i])")
+    scatter!(x, u_exact(x, t[i]),label="Exact, t=$(t[i])")
+end
+display(plt)
+```
+
+### Robin boundary conditions
+
+```julia
+using ModelingToolkit, DiffEqOperators 
+# Method of Manufactured Solutions
+u_exact = (x,t) -> exp.(-t) * sin.(x)
+
+# Parameters, variables, and derivatives
+@parameters t x
+@variables u(..)
+@derivatives Dt'~t
+@derivatives Dx'~x
+@derivatives Dxx''~x
+
+# 1D PDE and boundary conditions
+eq  = Dt(u(t,x)) ~ Dxx(u(t,x))
+bcs = [u(0,x) ~ sin(x),
+        u(t,-1.0) + 3Dx(u(t,-1.0)) ~ exp(-t) * (sin(-1.0) + 3cos(-1.0)),
+        u(t,1.0) + Dx(u(t,1.0)) ~ exp(-t) * (sin(1.0) + cos(1.0))]
+
+# Space and time domains
+domains = [t ∈ IntervalDomain(0.0,1.0),
+        x ∈ IntervalDomain(-1.0,1.0)]
+
+# PDE system
+pdesys = PDESystem(eq,bcs,domains,[t,x],[u])
+
+# Method of lines discretization
+# Need a small dx here for accuracy
+dx = 0.05
+order = 2
+discretization = MOLFiniteDifference(dx,order)
+
+# Convert the PDE problem into an ODE problem
+prob = discretize(pdesys,discretization)
+
+# Solve ODE problem
+using OrdinaryDiffEq
+sol = solve(prob,Tsit5(),saveat=0.2)
+
+# Plot results and compare with exact solution
+x = prob.space[2]
+t = sol.t
+
+using Plots
+plt = plot()
+for i in 1:length(t)
+    plot!(x,Array(prob.extrapolation[1](t[i])*sol.u[i]),label="Numerical, t=$(t[i])")
+    scatter!(x, u_exact(x, t[i]),label="Exact, t=$(t[i])")
+end
+display(plt)
 ```

src/DiffEqOperators.jl

@@ -21,6 +21,9 @@ include("jacvec_operators.jl")
 ### Utilities
 include("utils.jl")
 
+### Exceptions
+include("exceptions.jl")
+
 ### Boundary Padded Arrays
 include("boundary_padded_arrays.jl")
 
@@ -63,4 +66,5 @@ export compose, decompose, perpsize
 
 export GhostDerivativeOperator
 export MOLFiniteDifference
+export BoundaryConditionError
 end # module