add second order check

Author: ChrisRackauckas

test/2nd_order_check.jl

@@ -0,0 +1,51 @@
+using DiffEqOperators, Base.Test
+
+ord = 2               # Order of approximation
+Δx = 0.025π
+N = Int(2*(π/Δx)) -2
+x = -π:Δx:π-Δx
+D1  = DerivativeOperator{Float64}(1,ord,Δx,N,:periodic,:periodic);    # 1nd Derivative
+D2  = DerivativeOperator{Float64}(2,ord,Δx,N,:periodic,:periodic);    # 2nd Derivative
+
+u0 = cos(x)
+du_true = -cos(x)
+
+M = full(Tridiagonal([1.0 for i in 1:N+1],[-2.0 for i in 1:N+2],[1.0 for i in 1:N+1]))
+# Do the reflections, different for x and y operators
+M[end,1] = 1.0
+M[1,end] = 1.0
+M = M/(Δx^2)
+@test M*u0 ≈ D2*u0
+
+A = zeros(M)
+for i in 1:N+2, j in 1:N+2
+    i == j && (A[i,j] = -0.5)
+    abs(i - j) == 2 && (A[i,j] = 0.25)
+end
+A[end-1,1] = 0.25
+A[end,2] = 0.25
+A[1,end-1] = 0.25
+A[2,end] = 0.25
+A = A/(Δx^2)
+
+@test A*u0 ≈ D1*(D1*u0)
+
+κ(x) = 1.0
+const tmp1 = similar(x)
+const tmp2 = similar(x)
+function f(t,u,du)
+    A_mul_B!(tmp1,  D1, u)
+    @. tmp2 = κ(x)*tmp1
+    A_mul_B!(du,D1,tmp2)
+end
+
+du = similar(u0)
+f(0,u0,du)
+du ≈ A*u0
+
+error = du_true - du
+du2 = D2*u0
+error2 = du_true - du2
+
+@test maximum(error) < 0.004
+@test maximum(error2) < 0.001

test/runtests.jl

@@ -7,6 +7,7 @@ tic()
 @time @testset "Dirichlet BCs" begin include("dirichlet.jl") end
 @time @testset "Periodic BCs" begin include("periodic.jl") end
 @time @testset "Neumann BCs" begin include("neumann.jl") end
+@time @testset "2nd order check" begin include("2nd_order_check.jl") end
 @time @testset "None BCs" begin include("none.jl") end
 @time @testset "KdV" begin include("KdV.jl") end # KdV times out
 @time @testset "Heat Equation" begin include("heat_eqn.jl") end