analysis_dg1d.jl

# By default, Julia/LLVM does not use fused multiply-add operations (FMAs).
# Since these FMAs can increase the performance of many numerical algorithms,
# we need to opt-in explicitly.
# See https://ranocha.de/blog/Optimizing_EC_Trixi for further details.
@muladd begin
#! format: noindent

function create_cache_analysis(analyzer, mesh::TreeMesh{1},
                               equations, dg::DG, cache,
                               RealT, uEltype)

    # pre-allocate buffers
    # We use `StrideArray`s here since these buffers are used in performance-critical
    # places and the additional information passed to the compiler makes them faster
    # than native `Array`s.
    u_local = StrideArray(undef, uEltype,
                          StaticInt(nvariables(equations)), StaticInt(nnodes(analyzer)))
    x_local = StrideArray(undef, RealT,
                          StaticInt(ndims(equations)), StaticInt(nnodes(analyzer)))

    return (; u_local, x_local)
end

function create_cache_analysis(analyzer, mesh::StructuredMesh{1},
                               equations, dg::DG, cache,
                               RealT, uEltype)

    # pre-allocate buffers
    # We use `StrideArray`s here since these buffers are used in performance-critical
    # places and the additional information passed to the compiler makes them faster
    # than native `Array`s.
    u_local = StrideArray(undef, uEltype,
                          StaticInt(nvariables(equations)), StaticInt(nnodes(analyzer)))
    x_local = StrideArray(undef, RealT,
                          StaticInt(ndims(equations)), StaticInt(nnodes(analyzer)))
    jacobian_local = StrideArray(undef, RealT,
                                 StaticInt(nnodes(analyzer)))

    return (; u_local, x_local, jacobian_local)
end

function calc_error_norms(func, u, t, analyzer,
                          mesh::TreeMesh{1}, equations, initial_condition,
                          dg::DGSEM, cache, cache_analysis)
    @unpack vandermonde, weights = analyzer
    @unpack node_coordinates = cache.elements
    @unpack u_local, x_local = cache_analysis

    # Set up data structures
    l2_error = zero(func(get_node_vars(u, equations, dg, 1, 1), equations))
    linf_error = copy(l2_error)

    # Iterate over all elements for error calculations
    for element in eachelement(dg, cache)
        # Interpolate solution and node locations to analysis nodes
        multiply_dimensionwise!(u_local, vandermonde, view(u, :, :, element))
        multiply_dimensionwise!(x_local, vandermonde,
                                view(node_coordinates, :, :, element))

        # Calculate errors at each analysis node
        volume_jacobian_ = volume_jacobian(element, mesh, cache)

        for i in eachnode(analyzer)
            u_exact = initial_condition(get_node_coords(x_local, equations, dg, i), t,
                                        equations)
            diff = func(u_exact, equations) -
                   func(get_node_vars(u_local, equations, dg, i), equations)
            l2_error += diff .^ 2 * (weights[i] * volume_jacobian_)
            linf_error = @. max(linf_error, abs(diff))
        end
    end

    # For L2 error, divide by total volume
    total_volume_ = total_volume(mesh)
    l2_error = @. sqrt(l2_error / total_volume_)

    return l2_error, linf_error
end

function calc_error_norms(func, u, t, analyzer,
                          mesh::StructuredMesh{1}, equations, initial_condition,
                          dg::DGSEM, cache, cache_analysis)
    @unpack vandermonde, weights = analyzer
    @unpack node_coordinates, inverse_jacobian = cache.elements
    @unpack u_local, x_local, jacobian_local = cache_analysis

    # Set up data structures
    l2_error = zero(func(get_node_vars(u, equations, dg, 1, 1), equations))
    linf_error = copy(l2_error)
    total_volume = zero(real(mesh))

    # Iterate over all elements for error calculations
    for element in eachelement(dg, cache)
        # Interpolate solution and node locations to analysis nodes
        multiply_dimensionwise!(u_local, vandermonde, view(u, :, :, element))
        multiply_dimensionwise!(x_local, vandermonde,
                                view(node_coordinates, :, :, element))
        multiply_scalar_dimensionwise!(jacobian_local, vandermonde,
                                       inv.(view(inverse_jacobian, :, element)))

        # Calculate errors at each analysis node
        for i in eachnode(analyzer)
            u_exact = initial_condition(get_node_coords(x_local, equations, dg, i), t,
                                        equations)
            diff = func(u_exact, equations) -
                   func(get_node_vars(u_local, equations, dg, i), equations)
            # We take absolute value as we need the Jacobian here for the volume calculation
            abs_jacobian_local_i = abs(jacobian_local[i])

            l2_error += diff .^ 2 * (weights[i] * abs_jacobian_local_i)
            linf_error = @. max(linf_error, abs(diff))
            total_volume += weights[i] * abs_jacobian_local_i
        end
    end

    # For L2 error, divide by total volume
    l2_error = @. sqrt(l2_error / total_volume)

    return l2_error, linf_error
end

# Use quadrature to numerically integrate a single element.
# We do not multiply by the Jacobian to stay in reference space.
# This avoids the need to divide the RHS of the DG scheme by the Jacobian when computing
# the time derivative of entropy, see `entropy_change_reference_element`.
function integrate_reference_element(func::Func, u, element,
                                     ::Type{<:AbstractMesh{1}}, equations, dg::DGSEM,
                                     cache,
                                     args...) where {Func}
    @unpack weights = dg.basis

    # Initialize integral with zeros of the right shape
    element_integral = zero(func(u, 1, element, equations, dg, args...))

    for i in eachnode(dg)
        element_integral += weights[i] *
                            func(u, i, element, equations, dg, args...)
    end

    return element_integral
end

# Calculate ∫_e (∂S/∂u ⋅ ∂u/∂t) dΩ_e where the result on element 'e' is kept in reference space
# Note that ∂S/∂u = w(u) with entropy variables w
function entropy_change_reference_element(du, u, element,
                                          MeshT::Type{<:AbstractMesh{1}},
                                          equations, dg::DGSEM, cache, args...)
    return integrate_reference_element(u, element, MeshT, equations, dg, cache,
                                       du) do u, i, element, equations, dg, du
        u_node = get_node_vars(u, equations, dg, i, element)
        du_node = get_node_vars(du, equations, dg, i, element)

        dot(cons2entropy(u_node, equations), du_node)
    end
end

# calculate surface integral of func(u, equations) * normal on the reference element.
function surface_integral_reference_element(func::Func, u, element,
                                            ::Type{<:Union{TreeMesh{1},
                                                           StructuredMesh{1}}},
                                            equations, dg::DGSEM,
                                            cache, args...) where {Func}
    u_left = get_node_vars(u, equations, dg, 1, element)
    u_right = get_node_vars(u, equations, dg, nnodes(dg), element)

    surface_integral = func(u_right, 1, equations) - func(u_left, 1, equations)
    return surface_integral
end

function integrate_via_indices(func::Func, u,
                               mesh::TreeMesh{1}, equations, dg::DGSEM, cache,
                               args...; normalize = true) where {Func}
    @unpack weights = dg.basis

    # Initialize integral with zeros of the right shape
    integral = zero(func(u, 1, 1, equations, dg, args...))

    # Use quadrature to numerically integrate over entire domain
    @batch reduction=(+, integral) for element in eachelement(dg, cache)
        volume_jacobian_ = volume_jacobian(element, mesh, cache)
        for i in eachnode(dg)
            integral += volume_jacobian_ * weights[i] *
                        func(u, i, element, equations, dg, args...)
        end
    end

    # Normalize with total volume
    if normalize
        integral = integral / total_volume(mesh)
    end

    return integral
end

function integrate_via_indices(func::Func, u,
                               mesh::StructuredMesh{1}, equations, dg::DGSEM, cache,
                               args...; normalize = true) where {Func}
    @unpack weights = dg.basis

    # Initialize integral with zeros of the right shape
    integral = zero(func(u, 1, 1, equations, dg, args...))
    total_volume = zero(real(mesh))

    # Use quadrature to numerically integrate over entire domain
    @batch reduction=((+, integral), (+, total_volume)) for element in eachelement(dg,
                                                                                   cache)
        for i in eachnode(dg)
            jacobian_volume = abs(inv(cache.elements.inverse_jacobian[i, element]))
            integral += jacobian_volume * weights[i] *
                        func(u, i, element, equations, dg, args...)
            total_volume += jacobian_volume * weights[i]
        end
    end
    # Normalize with total volume
    if normalize
        integral = integral / total_volume
    end

    return integral
end

function integrate(func::Func, u,
                   mesh::Union{TreeMesh{1}, StructuredMesh{1}},
                   equations, dg::Union{DGSEM, FDSBP}, cache;
                   normalize = true) where {Func}
    integrate_via_indices(u, mesh, equations, dg, cache;
                          normalize = normalize) do u, i, element, equations, dg
        u_local = get_node_vars(u, equations, dg, i, element)
        return func(u_local, equations)
    end
end

function analyze(::typeof(entropy_timederivative), du, u, t,
                 mesh::Union{TreeMesh{1}, StructuredMesh{1}}, equations,
                 dg::Union{DGSEM, FDSBP}, cache)
    # Calculate ∫(∂S/∂u ⋅ ∂u/∂t)dΩ
    integrate_via_indices(u, mesh, equations, dg, cache,
                          du) do u, i, element, equations, dg, du
        u_node = get_node_vars(u, equations, dg, i, element)
        du_node = get_node_vars(du, equations, dg, i, element)
        return dot(cons2entropy(u_node, equations), du_node)
    end
end

function analyze(::Val{:l2_divb}, du, u, t,
                 mesh::TreeMesh{1}, equations::IdealGlmMhdEquations1D,
                 dg::DGSEM, cache)
    integrate_via_indices(u, mesh, equations, dg, cache,
                          dg.basis.derivative_matrix) do u, i, element, equations, dg,
                                                         derivative_matrix
        divb = zero(eltype(u))
        for k in eachnode(dg)
            divb += derivative_matrix[i, k] * u[6, k, element]
        end
        divb *= cache.elements.inverse_jacobian[element]
        return divb^2
    end |> sqrt
end

function analyze(::Val{:linf_divb}, du, u, t,
                 mesh::TreeMesh{1}, equations::IdealGlmMhdEquations1D,
                 dg::DGSEM, cache)
    @unpack derivative_matrix, weights = dg.basis

    # integrate over all elements to get the divergence-free condition errors
    linf_divb = zero(eltype(u))
    @batch reduction=(max, linf_divb) for element in eachelement(dg, cache)
        for i in eachnode(dg)
            divb = zero(eltype(u))
            for k in eachnode(dg)
                divb += derivative_matrix[i, k] * u[6, k, element]
            end
            divb *= cache.elements.inverse_jacobian[element]
            linf_divb = max(linf_divb, abs(divb))
        end
    end

    return linf_divb
end
end # @muladd