--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/test/SbpOperators/volumeops/derivatives/second_derivative_test.jl	Thu Jan 27 10:55:08 2022 +0100
@@ -0,0 +1,118 @@
+using Test
+
+using Sbplib.SbpOperators
+using Sbplib.Grids
+using Sbplib.LazyTensors
+
+import Sbplib.SbpOperators.VolumeOperator
+
+@testset "SecondDerivative" begin
+    stencil_set = read_stencil_set(sbp_operators_path()*"standard_diagonal.toml"; order=4)
+    inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
+    closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
+    Lx = 3.5
+    Ly = 3.
+    g_1D = EquidistantGrid(121, 0.0, Lx)
+    g_2D = EquidistantGrid((121,123), (0.0, 0.0), (Lx, Ly))
+
+    @testset "Constructors" begin
+        @testset "1D" begin
+            Dₓₓ = second_derivative(g_1D,inner_stencil,closure_stencils)
+            @test Dₓₓ == second_derivative(g_1D,inner_stencil,closure_stencils,1)
+            @test Dₓₓ isa VolumeOperator
+        end
+        @testset "2D" begin
+            Dₓₓ = second_derivative(g_2D,inner_stencil,closure_stencils,1)
+            D2 = second_derivative(g_1D,inner_stencil,closure_stencils)
+            I = IdentityMapping{Float64}(size(g_2D)[2])
+            @test Dₓₓ == D2⊗I
+            @test Dₓₓ isa TensorMapping{T,2,2} where T
+        end
+    end
+
+    # Exact differentiation is measured point-wise. In other cases
+    # the error is measured in the l2-norm.
+    @testset "Accuracy" begin
+        @testset "1D" begin
+            l2(v) = sqrt(spacing(g_1D)[1]*sum(v.^2));
+            monomials = ()
+            maxOrder = 4;
+            for i = 0:maxOrder-1
+                f_i(x) = 1/factorial(i)*x^i
+                monomials = (monomials...,evalOn(g_1D,f_i))
+            end
+            v = evalOn(g_1D,x -> sin(x))
+            vₓₓ = evalOn(g_1D,x -> -sin(x))
+
+            # 2nd order interior stencil, 1nd order boundary stencil,
+            # implies that L*v should be exact for monomials up to order 2.
+            @testset "2nd order" begin
+                stencil_set = read_stencil_set(sbp_operators_path()*"standard_diagonal.toml"; order=2)
+                inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
+			    closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
+                Dₓₓ = second_derivative(g_1D,inner_stencil,closure_stencils)
+                @test Dₓₓ*monomials[1] ≈ zeros(Float64,size(g_1D)...) atol = 5e-10
+                @test Dₓₓ*monomials[2] ≈ zeros(Float64,size(g_1D)...) atol = 5e-10
+                @test Dₓₓ*monomials[3] ≈ monomials[1] atol = 5e-10
+                @test Dₓₓ*v ≈ vₓₓ rtol = 5e-2 norm = l2
+            end
+
+            # 4th order interior stencil, 2nd order boundary stencil,
+            # implies that L*v should be exact for monomials up to order 3.
+            @testset "4th order" begin
+                stencil_set = read_stencil_set(sbp_operators_path()*"standard_diagonal.toml"; order=4)
+                inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
+			    closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
+                Dₓₓ = second_derivative(g_1D,inner_stencil,closure_stencils)
+                # NOTE: high tolerances for checking the "exact" differentiation
+                # due to accumulation of round-off errors/cancellation errors?
+                @test Dₓₓ*monomials[1] ≈ zeros(Float64,size(g_1D)...) atol = 5e-10
+                @test Dₓₓ*monomials[2] ≈ zeros(Float64,size(g_1D)...) atol = 5e-10
+                @test Dₓₓ*monomials[3] ≈ monomials[1] atol = 5e-10
+                @test Dₓₓ*monomials[4] ≈ monomials[2] atol = 5e-10
+                @test Dₓₓ*v ≈ vₓₓ rtol = 5e-4 norm = l2
+            end
+        end
+
+        @testset "2D" begin
+            l2(v) = sqrt(prod(spacing(g_2D))*sum(v.^2));
+            binomials = ()
+            maxOrder = 4;
+            for i = 0:maxOrder-1
+                f_i(x,y) = 1/factorial(i)*y^i + x^i
+                binomials = (binomials...,evalOn(g_2D,f_i))
+            end
+            v = evalOn(g_2D, (x,y) -> sin(x)+cos(y))
+            v_yy = evalOn(g_2D,(x,y) -> -cos(y))
+
+            # 2nd order interior stencil, 1st order boundary stencil,
+            # implies that L*v should be exact for binomials up to order 2.
+            @testset "2nd order" begin
+                stencil_set = read_stencil_set(sbp_operators_path()*"standard_diagonal.toml"; order=2)
+                inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
+                closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
+                Dyy = second_derivative(g_2D,inner_stencil,closure_stencils,2)
+                @test Dyy*binomials[1] ≈ zeros(Float64,size(g_2D)...) atol = 5e-9
+                @test Dyy*binomials[2] ≈ zeros(Float64,size(g_2D)...) atol = 5e-9
+                @test Dyy*binomials[3] ≈ evalOn(g_2D,(x,y)->1.) atol = 5e-9
+                @test Dyy*v ≈ v_yy rtol = 5e-2 norm = l2
+            end
+
+            # 4th order interior stencil, 2nd order boundary stencil,
+            # implies that L*v should be exact for binomials up to order 3.
+            @testset "4th order" begin
+                stencil_set = read_stencil_set(sbp_operators_path()*"standard_diagonal.toml"; order=4)
+                inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
+                closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
+                Dyy = second_derivative(g_2D,inner_stencil,closure_stencils,2)
+                # NOTE: high tolerances for checking the "exact" differentiation
+                # due to accumulation of round-off errors/cancellation errors?
+                @test Dyy*binomials[1] ≈ zeros(Float64,size(g_2D)...) atol = 5e-9
+                @test Dyy*binomials[2] ≈ zeros(Float64,size(g_2D)...) atol = 5e-9
+                @test Dyy*binomials[3] ≈ evalOn(g_2D,(x,y)->1.) atol = 5e-9
+                @test Dyy*binomials[4] ≈ evalOn(g_2D,(x,y)->y) atol = 5e-9
+                @test Dyy*v ≈ v_yy rtol = 5e-4 norm = l2
+            end
+        end
+    end
+end