From a940bcc6405f6e1be4dd301d9337cbdc9d9c55de Mon Sep 17 00:00:00 2001
From: Florian Atteneder <florian.atteneder@uni-jena.de>
Date: Tue, 13 Aug 2024 13:51:06 +0000
Subject: [PATCH] dg1d: refactor lgl and glgl modules
 (https://git.tpi.uni-jena.de/dg/dg1d.jl/-/merge_requests/204)

Unify interface for LGL and GLGL rules and update `SpectralElement` accordingly:
- most routine should just work with one argument `Npts`
- add more doc strings
- remove unused methods `linear_dependency_matrix` and `expansion_matrix`
- remove unused fields of `SpectralElement`
---
 src/glgl.jl                                   | 235 +++++-------------
 src/lgl.jl                                    | 192 +++-----------
 src/spectralelement.jl                        |  52 ++--
 .../refs/burgers_sine_avmda/output1d.h5       | Bin 40984 -> 40984 bytes
 .../output1d.h5                               | Bin 41784 -> 41784 bytes
 .../refs/grhd_bondi_accretion/output0d.h5     | Bin 25300 -> 25300 bytes
 .../refs/grhd_bondi_accretion/output1d.h5     | Bin 182104 -> 182104 bytes
 .../refs/grhd_bondi_infall_avmda/output1d.h5  | Bin 60280 -> 60280 bytes
 .../refs/srhd_simple_wave_avmda/output1d.h5   | Bin 122840 -> 122840 bytes
 test/UnitTests/src/test_spectral.jl           | 118 +++++----
 10 files changed, 193 insertions(+), 404 deletions(-)

diff --git a/src/glgl.jl b/src/glgl.jl
index 47d47cd5..54207a06 100644
--- a/src/glgl.jl
+++ b/src/glgl.jl
@@ -1,22 +1,23 @@
 module GLGL
 
 
-using LinearAlgebra, Jacobi, ToggleableAsserts
-import dg1d: purge_zeros!, lagrange_poly
+using Jacobi, LinearAlgebra, ToggleableAsserts
+import dg1d: purge_zeros!, lagrange_poly, first_derivative_matrix, second_derivative_matrix
 
 
 """
-Computes quadrature nodes, weights and derivative weights
-for the Generalized-Gauss-Lobatto quadrature rule.
-n_pts ... number of quadrature nodes
-Return values
-x ... quadrature nodes, vector
-w ... quadrature weights, vector
-v ... derivative weight, number
+  rule(Npts::Integer)
+
+Computes quadrature nodes and weights for the Generalized-Legendre-Gauss-Lobatto quadrature rule.
 """
-function rule(n_pts::Integer)
+function rule(Npts::Integer)
+  x, w, v, D = inner_product_rule(Npts)
+  @. w += v * (D[1,:] - D[end,:])
+  return x, w
+end
+function inner_product_rule(Npts::Integer)
+  N = Npts - 2
 
-  N = n_pts - 2
   gb_x, gb_w = [], []
   if N > 0
     # Integration nodes and weights for Gauss-Gegenbauer quadrature
@@ -24,229 +25,127 @@ function rule(n_pts::Integer)
     gb_x = zgj(N,    α, β)
     gb_w = wgj(gb_x, α, β)
   end
-
-  x = zeros(n_pts)
-  w = zeros(n_pts)
+  x = zeros(Npts)
+  w = zeros(Npts)
 
   x[2:end-1] .= gb_x
   x[1] = -1.0
   x[end] = 1.0
-
   w[2:end-1] .= @. gb_w / (1.0 - gb_x^2)^2
-
   w[1] = w[end] = 8.0 / 3.0 * (2.0 * N^2 + 10.0 * N + 9.0) / ((N+1.0) * (N+2.0) * (N+3.0) * (N+4.0))
   v = 8.0 / ((N+1.0) * (N+2.0) * (N+3.0) * (N+4.0))
 
-  return x, w, v
-end # function
-
-
-
-"""
-  integrate(f, x, w, v, D)
-
-Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`,
-boundary weight `v` and derivative matrix `D` based on `x`.
-Use `rule` to obtain `x, w, v` and use `first_derivative_matrix` to obtain `D`.
-"""
-function integrate(f::AbstractVector, x::AbstractVector, w::AbstractVector, v::Real, D::AbstractVector)
-
-  N = length(w)
-  @toggled_assert (N,N) == size(D)
-  @toggled_assert N == length(f)
-
-  sum = dot(f, w)
-  sum += v * dot(view(D,1,:), f)
-  sum -= v * dot(view(D,N,:), f)
+  D = first_derivative_matrix(x)
 
-  return sum
+  return x, w, v, D
 end
 
 
 """
-  integrate(f1, f2, w, v, D)
+  integrate(f, x, w)
 
-Similar to `integrate(f, w)`, but computes the inner product of `f1*f2` where both
-`f1, f2` must be sampled on the GLGL quadrature nodes `x` associated with the quadrature
-weights `w`, boundary weight `v` and derivative matrix `D` based on `x`.
-Use `rule` to obtain `x, w, v` and use `first_derivative_matrix` to obtain `D`.
+Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`.
+Use `rule` to obtain `x, w`.
 """
-function integrate(f1::AbstractVector, f2::AbstractVector, x::AbstractVector, w::AbstractVector,
-    v::Real, D::AbstractMatrix)
-
-  N = length(w)
-  @toggled_assert (N,N) == size(D)
-  @toggled_assert N == length(f1)
-  @toggled_assert N == length(f2)
-
-  sum = dot(f1.*f2, w)
-  vD1, vDN = view(D,1,:), view(D,N,:)
-  sum += v * (f1[1] * dot(vD1, f2) + f2[1] * dot(vD1, f1))
-  sum -= v * (f1[end] * dot(vDN, f2) + f2[end] * dot(vDN, f1))
-
-  return sum
+function integrate(f::AbstractVector, x::AbstractVector, w::AbstractVector)
+  @toggled_assert length(w) == length(f) == length(x)
+  return dot(w, f)
 end
 
 
 """
   integrate(f::Function, idxs::AbstractVector{<:Integer}, x, w)
 
-Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`,
-boundary weight `v` and derivative matrix `D` based on `x`.
+Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`.
 `f(idxs[i])` shall return the value of the integrand evaluated at `x[i]`.
-Use `rule` to obtain `x, w, v` and use `first_derivative_matrix` to obtain `D`.
+Use `rule` to obtain `x, w`.
 """
-function integrate(fn::Function, idxs::AbstractVector{<:Integer}, x::AbstractVector, w::AbstractVector,
-    v::Real, D::AbstractMatrix)
-
-  N = length(w)
-  @toggled_assert (N,N) == size(D)
-  @toggled_assert N == length(idxs)
-
+function integrate(fn::Function, idxs::AbstractVector{<:Integer}, x::AbstractVector)
+  @toggled_assert length(w) == length(idxs)
   sum = 0.0
-  for i in 1:N
-    fi = fn(idxs[i])
-    sum += (w[i] + v * (D[1,i] - D[N,i])) * fi
+  for i in 1:length(w)
+    sum += w[i] * f(idxs[i])
   end
-
   return sum
 end
 
 
 """
-  integrate(f::Function, x, w, v, D)
+  integrate(f::Function, x, w)
 
-Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`,
-boundary weight `v` and derivative matrix `D` based on `x`.
+Integrate `f` sampled on GLGL quadrature nodes `x` with quadrature weights `w`.
 `f(x[i])` shall return the value of the integrand evaluated at `x[i]`.
-Use `rule` to obtain `x, w, v` and use `first_derivative_matrix` to obtain `D`.
+Use `rule` to obtain `x, w`.
 
-This is similar to `integrate(f, idxs, x, w, v, D)`, but is better suited if the integrand is
+This is similar to `integrate(f, idxs, x, w)`, but is better suited if the integrand is
 given in closed form.
 """
-function integrate(f::Function, x::AbstractVector, w::AbstractVector, v::Real, D::AbstractMatrix)
-  @toggled_assert length(w) == length(x) == size(D, 1) == size(D, 2)
+function integrate(f::Function, x::AbstractVector, w::AbstractVector)
+  @toggled_assert length(w) == length(x)
   sum = 0.0
   N = length(w)
   for i in 1:N
     fi = f(x[i])
-    sum += (w[i] + v * (D[1,i] - D[N,i])) * fi
+    sum += w[i] * fi
   end
   return sum
 end
 
 
 """
-M_ij = (li, lj)
-x ... GLGL nodes
-w ... GLGL weights
-v ... GLGL derivative weight
-D ... Derivative matrix based on x.
+  inner_product(f1, f2, x, w)
+
+Computed the inner product between `f1` and `f2` sampled on GLGL quadrature nodes `x`
+with quadrature weights `w`, boundary weights `v` and first derivative matrix `D`
+Use `inner_product_rule` to obtain `x, w, v, D`.
 """
-function mass_matrix(x::AbstractVector, w::AbstractVector, v::Real, D::AbstractMatrix)
+function inner_product(f1::AbstractVector, f2::AbstractVector, x::AbstractVector, w::AbstractVector,
+    v::Real, D::AbstractMatrix)
+  N = length(w)
+  @toggled_assert (N,N) == size(D)
+  @toggled_assert N == length(f1)
+  @toggled_assert N == length(f2)
+  sum = dot(f1.*f2, w)
+  vD1, vDN = view(D,1,:), view(D,N,:)
+  sum += v * (f1[1] * dot(vD1, f2) + f2[1] * dot(vD1, f1))
+  sum -= v * (f1[end] * dot(vDN, f2) + f2[end] * dot(vDN, f1))
+  return sum
+end
 
-  N = length(x)
-  @assert N == length(w)
-  @assert (N,N) == size(D)
 
+"""
+  mass_matrix(Npts::Integer)
+
+Mass matrix for Lagrange polynomials based on GLGL quadrature nodes.
+M_ij = (li, lj)
+"""
+function mass_matrix(Npts::Integer)
+  x, w, v, D = inner_product_rule(Npts)
   M = diagm(w)
   M[1,:] += v * D[1,:]
   M[:,1] += v * D[1,:]
   M[end,:] -= v * D[end,:]
   M[:,end] -= v * D[end,:]
-
   return M
-end # function
-
+end
 
 
 """
+  stiffness_matrix(Npts::Integer)
+
+Stiffness matrix for Lagrange polynomials based on GLGL quadrature nodes.
 S_ij = (li, dlj/dx)
-Exactly representable, because product is of order 2N - 1 which can be integrated
-exactly using LGL quadrature.
-x ... GLGL nodes
-w ... GLGL weights
-v ... GLGL derivative weight
-D ... First derivative matrix based on x.
-D2 ... Second derivative matrix based on x.
 """
-function stiffness_matrix(x::AbstractVector, w::AbstractVector, v::Real,
-    D::AbstractMatrix, D2::AbstractMatrix)
-
-  N = length(x)
-  @assert N == length(w)
-  @assert (N,N) == size(D)
-  @assert (N,N) == size(D2)
-
+function stiffness_matrix(Npts::Integer)
+  x, w, v, D = inner_product_rule(Npts)
+  D2 = second_derivative_matrix(x)
   S = w .* D
   S[1,:]    += v * D2[1,:]
   S[end,:]  -= v * D2[end,:]
   S .+= v * D[1,:] .* transpose(D[1,:])
   S .-= v * D[end,:] .* transpose(D[end,:])
-
   return S
 end
 
 
-# TODO Add versions of mass_matrix, stiffness_matrix that take
-# a single argument n_pts.
-
-
-"""
-L_ij = (P_i, P_j)
-Analogon to mass matrix of Lagrange polynomials, expresses the linear dependency
-of the numerical representation of the Legendre basis. Ideally, it is a
-diagonal matrix (which should be ensured by the GLGL quadrature).
-x ... GLGL nodes
-w ... GLGL weights
-v ... GLGL derivative weight
-D ... First derivative matrix based on x.
-"""
-function linear_dependency_matrix(x::AbstractVector, w::AbstractVector, v::Real, D::AbstractMatrix)
-  N = length(x)
-  L = zeros(N,N)
-  for i in 1:N
-    Pi = map(x->legendre(x, i-1), x)
-    for j in i:N
-      Pj = map(x->legendre(x, j-1), x)
-      lij = integrate(Pi, Pj, x, w, v, D)
-      L[i,j] = lij
-      if i != j
-        L[j,i] = L[i,j]
-      end
-    end
-  end
-  purge_zeros!(L)
-
-  return L
 end
-
-
-"""
-E_ij = (l_i, P_j)
-Commonly used to project either Lagrange polynomials onto Legendres or
-reverse.
-x ... GLGL nodes
-w ... GLGL weights
-v ... GLGL derivative weight
-D ... First derivative matrix based on x.
-"""
-function expansion_matrix(x::AbstractVector, w::AbstractVector, v::Real, D::AbstractMatrix)
-  N = length(x)
-  T = zeros(N,N)
-  for i in 1:N
-    li = map(xi->lagrange_poly(i, xi, x), x)
-    for j in 1:N
-      Pj = map(x->legendre(x, j-1), x)
-      tij = integrate(li, Pj, x, w, v, D)
-      T[i,j] = tij
-    end
-  end
-  purge_zeros!(T)
-
-  return T
-end
-
-
-
-end # module
diff --git a/src/lgl.jl b/src/lgl.jl
index 31309a41..6a7c97e8 100644
--- a/src/lgl.jl
+++ b/src/lgl.jl
@@ -5,21 +5,19 @@ using Jacobi, LinearAlgebra, ToggleableAsserts
 import dg1d: lagrange_poly, first_derivative_matrix, purge_zeros!
 
 
-
 """
-Quadrature nodes and weights for Gauss-Legendre rule.
-n_pts ... number of quadrature points
-Exact up to order 2*n_pts - 1.
+  rule(Npts::Integer)
+
+Computes quadrature nodes and weights for the Legendre-Gauss-Lobatto-Legendre quadrature rule.
 """
-function rule(n_pts::Integer)
+function rule(Npts::Integer)
   α = 0.0; β = 0.0
-  x = zglj(n_pts, α, β)
+  x = zglj(Npts, α, β)
   w = wglj(x, α, β)
   return x, w
 end
 
 
-
 """
   integrate(f, x, w)
 
@@ -27,30 +25,11 @@ Integrate `f` sampled on LGL quadrature nodes `x` with quadrature weights `w`.
 Use `rule` to obtain `x, w`.
 """
 function integrate(f::AbstractVector, x::AbstractVector, w::AbstractVector)
-  @toggled_assert length(w) == length(f)
+  @toggled_assert length(w) == length(f) == length(x)
   return dot(w, f)
 end
 
 
-"""
-  integrate(f1, f2, x, w)
-
-Similar to `integrate(f, w)`, but computes the inner product of `f1*f2` where both
-`f1, f2` must be sampled on the LGL quadrature nodes `x` associated with the quadrature
-weights `w`.
-Use `rule` to obtain `x, w`.
-"""
-function integrate(f1::AbstractVector, f2::AbstractVector, x::AbstractVector, w::AbstractVector)
-  @toggled_assert length(w) == length(f1)
-  @toggled_assert length(w) == length(f2)
-  I = 0.0
-  for i = 1:length(w)
-    I += w[i] * f1[i] * f2[i]
-  end
-  return I
-end
-
-
 """
   integrate(f::Function, idxs::AbstractVector{<:Integer}, x, w)
 
@@ -88,6 +67,23 @@ function integrate(f::Function, x::AbstractVector, w::AbstractVector)
 end
 
 
+"""
+  inner_product(f1, f2, x, w)
+
+Computed the inner product between `f1` and `f2` sampled on LGL quadrature nodes `x`
+with quadrature weights `w`.
+Use `rule` to obtain `x, w`.
+"""
+function inner_product(f1::AbstractVector, f2::AbstractVector, x::AbstractVector, w::AbstractVector)
+  @toggled_assert length(w) == length(f1) == length(f2) == length(x)
+  sum = 0.0
+  for i in 1:length(w)
+    sum += w[i] * f1[i] * f2[i]
+  end
+  return sum
+end
+
+
 """
 Inverse of exact full mass matrix, computed using Sherman-Morrison formula,
 cf. PhD thesis Marcus Bugner, Notes spectral elements Bernd Bruegmann,
@@ -105,176 +101,64 @@ function inverse_mass_matrix_exact(n_pts::Integer)
 end
 
 
-
 """
 n_pts ... number of qudrature nodes = number of lagrange polynomials
 """
-function inverse_mass_matrix_lumped(n_pts::Integer)
-  _, w = rule(n_pts)
+function inverse_mass_matrix_lumped(Npts::Integer)
+  _, w = rule(Npts)
   invM = diagm(1.0 ./ w)
   return invM
 end
 
 
-
 """
 M_ij = (li, lj)
 n_pts ... number of qudrature nodes = number of lagrange polynomials
 """
-function mass_matrix_exact(n_pts::Integer)
-  z, w = rule(n_pts)
-  M = zeros(n_pts, n_pts)
-  aux_z, aux_w = rule(n_pts+2)
+function mass_matrix_exact(Npts::Integer)
+  z, w = rule(Npts)
+  M = zeros(Npts, Npts)
+  aux_z, aux_w = rule(Npts+2)
   L(i, x) = lagrange_poly(i, x, z)
-  M = zeros(n_pts, n_pts)
-  for i=1:n_pts, j=1:n_pts
+  M = zeros(Npts, Npts)
+  for i=1:Npts, j=i:Npts
     li = L.(i, aux_z)
-    for j = i:n_pts
+    for j = i:Npts
       lj = L.(j, aux_z)
-      M[i,j] = integrate(li, lj, aux_w)
-      if (i != j)
+      M[i,j] = inner_product(li, lj, aux_z, aux_w)
+      if j > i
         M[j,i] = M[i,j]
       end
     end
   end
-
   return M
 end
 
 
-
 """
 M_ij = (li, lj)
 Approximated mass matrix due to under integration of LGL quadrature.
 n_pts ... number of qudrature nodes = number of lagrange polynomials
 """
-function mass_matrix_lumped(n_pts::Integer)
-  z, w = rule(n_pts)
+function mass_matrix_lumped(Npts::Integer)
+  z, w = rule(Npts)
   M = diagm(w)
   return M
 end
 
 
-
 """
 S_ij = (li, dlj/dx)
 Exactly representable, because product is of order 2N - 1 which can be integrated
 exactly using LGL quadrature.
 n_pts ... number of quadrature nodes = number of lagrange polynomials
 """
-function stiffness_matrix(n_pts::Integer)
-  z, w = rule(n_pts)
+function stiffness_matrix(Npts::Integer)
+  z, w = rule(Npts)
   D = first_derivative_matrix(z)
   S = w .* D
   return S
 end
 
 
-
-"""
-L_ij = (P_i, P_j)
-Analogon to mass matrix of Lagrange polynomials, expresses the linear dependency
-of the numerical representation of the Legendre basis.
-x ... GLGL nodes
-w ... GLGL weights
-"""
-function linear_dependency_matrix_lumped(x::AbstractVector, w::AbstractVector)
-  N = length(x)
-  L = zeros(N,N)
-  for i in 1:N
-    Pi = map(x->legendre(x, i-1), x)
-    for j in i:N
-      Pj = map(x->legendre(x, j-1), x)
-      lij = integrate(Pi, Pj, x, w)
-      L[i,j] = lij
-      if i != j
-        L[j,i] = L[i,j]
-      end
-    end
-  end
-  purge_zeros!(L)
-
-  return L
-end
-
-
-"""
-E_ij = (l_i, P_j)
-Commonly used to project either Lagrange polynomials onto Legendres or
-reverse.
-x ... GLGL nodes
-w ... GLGL weights
-"""
-function expansion_matrix_lumped(x::AbstractVector, w::AbstractVector)
-  N = length(x)
-  T = zeros(N,N)
-  for i in 1:N
-    li = map(xi->lagrange_poly(i, xi, x), x)
-    for j in 1:N
-      Pj = map(x->legendre(x, j-1), x)
-      tij = integrate(li, Pj, x, w)
-      T[i,j] = tij
-    end
-  end
-  purge_zeros!(T)
-
-  return T
-end
-
-
-"""
-L_ij = (P_i, P_j)
-Analogon to mass matrix of Lagrange polynomials, expresses the linear dependency
-of the numerical representation of the Legendre basis.
-x ... GLGL nodes
-w ... GLGL weights
-"""
-function linear_dependency_matrix_exact(x::AbstractVector, w::AbstractVector)
-  N = length(x)
-  N2 = N+2
-  x2, w2 = rule(N2)
-  L = zeros(N,N)
-  for i in 1:N
-    Pi = map(x->legendre(x, i-1), x2)
-    for j in i:N
-      Pj = map(x->legendre(x, j-1), x2)
-      lij = integrate(Pi, Pj, x2, w2)
-      L[i,j] = lij
-      if i != j
-        L[j,i] = L[i,j]
-      end
-    end
-  end
-  purge_zeros!(L)
-
-  return L
-end
-
-
-"""
-E_ij = (l_i, P_j)
-Commonly used to project either Lagrange polynomials onto Legendres or
-reverse.
-x ... GLGL nodes
-w ... GLGL weights
-"""
-function expansion_matrix_exact(x::AbstractVector, w::AbstractVector)
-  N = length(x)
-  N2 = N + 2
-  x2, w2 = rule(N2)
-  T = zeros(N,N)
-  for i in 1:N
-    li = map(xi->lagrange_poly(i, xi, x), x2)
-    for j in 1:N
-      Pj = map(x->legendre(x, j-1), x2)
-      tij = integrate(li, Pj, x2, w2)
-      T[i,j] = tij
-    end
-  end
-  purge_zeros!(T)
-
-  return T
 end
-
-
-end # module LGL
diff --git a/src/spectralelement.jl b/src/spectralelement.jl
index dcf0b418..81b56b75 100644
--- a/src/spectralelement.jl
+++ b/src/spectralelement.jl
@@ -3,10 +3,9 @@ Base.@kwdef struct SpectralElement
   Npts::Int64             # number of quadrature points = N + 1
   z::Vector{Float64}      # quadrature points
   w::Vector{Float64}      # quadrature weights
-  v::Float64              # derivative weight (only needed for GLGL quadrature rule)
-  bw::Vector{Float64}     # barycentric weights
   D::Matrix{Float64}      # derivative matrix
   S::Matrix{Float64}      # stiffness matrix
+  M::Matrix{Float64}      # mass matrix
   invM::Matrix{Float64}   # inverse mass matrix
   MDM::Matrix{Float64}    # = (M^-1 D^T M) = M^-1 S^T
   Ml_lhs::Vector{Float64} # = M^-1 l(-1)
@@ -16,7 +15,7 @@ Base.@kwdef struct SpectralElement
   quadr::Symbol           # :LGL, :LGL_mass_lumped, :GLGL
   F::Matrix{Float64}      # spectral filtering matrix
 
-  function SpectralElement(N::Int, quadr::Symbol;
+  function SpectralElement(N::Integer, quadr::Symbol;
       # filter parameters
       ηc::Float64=0.5, sh::Float64=16.0, α::Float64=36.0)
     Npts = N + 1
@@ -25,35 +24,26 @@ Base.@kwdef struct SpectralElement
       z, w        = LGL.rule(Npts)
       v           = -1.0 # unused here
       purge_zeros!(z) # ensures that the centered node for even N basis sits exactly at x = 0.
-      bw          = barycentric_weights(z)
       D           = first_derivative_matrix(z)
       S           = LGL.stiffness_matrix(Npts)
+      M           = LGL.mass_matrix_exact(Npts)
       invM        = LGL.inverse_mass_matrix_exact(Npts)
-      # TODO Figure out how much the aliasing error influences the accuracy of the inner product.
-      # Could be totally dealised using the Orszag 2/3rds rule.
-      L           = LGL.linear_dependency_matrix_exact(z, w)
-      E           = LGL.expansion_matrix_exact(z, w)
     elseif quadr === :LGL_lumped
       z, w        = LGL.rule(Npts)
       v           = -1.0 # unused here
       purge_zeros!(z) # ensures that the centered node for even N basis sits exactly at x = 0.
-      bw          = barycentric_weights(z)
       D           = first_derivative_matrix(z)
       S           = LGL.stiffness_matrix(Npts)
+      M           = LGL.mass_matrix_lumped(Npts)
       invM        = LGL.inverse_mass_matrix_lumped(Npts)
-      L           = LGL.linear_dependency_matrix_lumped(z, w)
-      E           = LGL.expansion_matrix_lumped(z, w)
     elseif quadr === :GLGL
-      z, w, v     = GLGL.rule(Npts)
+      z, w        = GLGL.rule(Npts)
       purge_zeros!(z) # ensures that the centered node for even N basis sits exactly at x = 0.
-      bw          = barycentric_weights(z)
       D           = first_derivative_matrix(z)
       D2          = second_derivative_matrix(z)
-      M           = GLGL.mass_matrix(z, w, v, D)
-      S           = GLGL.stiffness_matrix(z, w, v, D, D2)
+      M           = GLGL.mass_matrix(Npts)
+      S           = GLGL.stiffness_matrix(Npts)
       invM        = inv(M)
-      L           = GLGL.linear_dependency_matrix(z, w, v, D)
-      E           = GLGL.expansion_matrix(z, w, v, D)
     else
       error("Invalid quadrature rule specified! Use one of :LGL_lumped, :LGL, :GLGL.")
     end
@@ -72,14 +62,14 @@ Base.@kwdef struct SpectralElement
     Ml_lhs = invM * l_lhs
     Ml_rhs = invM * l_rhs
 
-    return new(N, Npts, z, w, v, bw, D, S, invM, MDM, Ml_lhs, Ml_rhs,
+    return new(N, Npts, z, w, D, S, M, invM, MDM, Ml_lhs, Ml_rhs,
                V, invV, quadr, F)
   end
 end
 
 
 # (5.16) from Hesthaven, Warburton 2007
-function spectral_filter(η, prms)
+function spectral_filter(η::Real, prms)
   ηc, sh, α = prms
   return if 0 ≤ η ≤ ηc
     1
@@ -89,24 +79,23 @@ function spectral_filter(η, prms)
 end
 
 
-@inline function inner_product(u1, u2, el::SpectralElement)
-  @unpack quadr = el
-  @toggled_assert quadr in (:LGL, :LGL_lumped, :GLGL)
-  if quadr === :LGL || quadr === :LGL_lumped
-    return LGL.integrate(u1, u2, el.z, el.w)
-  else # quadr === :GLGL
-    return GLGL.integrate(u1, u2, el.z, el.w, el.v, el.D)
+@inline function inner_product(u1::AbstractVector, u2::AbstractVector, el::SpectralElement)
+  @unpack Npts, M = el
+  result = 0.0
+  @turbo for i in 1:Npts, j in 1:Npts
+    result += u1[i] * M[i,j] * u2[j]
   end
+  return result
 end
 
 
-@inline function integrate(u, el::SpectralElement)
+@inline function integrate(u::AbstractVector, el::SpectralElement)
   @unpack quadr = el
   @toggled_assert quadr in (:LGL, :LGL_lumped, :GLGL)
   if quadr === :LGL || quadr === :LGL_lumped
     return LGL.integrate(u, el.z, el.w)
   else # quadr === :GLGL
-    return GLGL.integrate(u, el.z, el.w, el.v, el.D)
+    return GLGL.integrate(u, el.z, el.w)
   end
 end
 
@@ -117,15 +106,16 @@ end
   if quadr === :LGL || quadr === :LGL_lumped
     return LGL.integrate(fn, idxs, el.z, el.w)
   else # quadr === :GLGL
-    return GLGL.integrate(fn, idxs, el.z, el.w, el.v, el.D)
+    return GLGL.integrate(fn, idxs, el.z, el.w)
   end
 end
 
 
-@inline function differentiate(u, el::SpectralElement)
+@inline function differentiate(u::AbstractArray, el::SpectralElement)
   return el.D * u
 end
 
-@inline function differentiate!(du, u, el::SpectralElement)
+
+@inline function differentiate!(du::AbstractArray, u::AbstractArray, el::SpectralElement)
   mul!(du, el.D, u)
 end
diff --git a/test/IntegrationTests/refs/burgers_sine_avmda/output1d.h5 b/test/IntegrationTests/refs/burgers_sine_avmda/output1d.h5
index d2e27152e4d1ac944e2d38cf87060dd1e7100f3f..1a041f434509447ed292d10e1db86b39f72d8094 100644
GIT binary patch
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diff --git a/test/IntegrationTests/refs/euler_sod_shock_tube_bernstein/output1d.h5 b/test/IntegrationTests/refs/euler_sod_shock_tube_bernstein/output1d.h5
index 7d2cb4eb0cb6a458e63086ae2c59b38af0b3f9e4..387eebaa35970d5fb6599bdae781512889b74e08 100644
GIT binary patch
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diff --git a/test/IntegrationTests/refs/grhd_bondi_accretion/output0d.h5 b/test/IntegrationTests/refs/grhd_bondi_accretion/output0d.h5
index 2c258cd6e73e717289c01b4a4a953373f070538d..add2a996056d73e37a32e3de6aa665dd59fc14b9 100644
GIT binary patch
delta 687
zcmca|l<~?@#tmE8n1#c3Pd@0~IC+7%&E^vJ1)R)MK!Jr`zb6X`DS_D_$qfowla++Q
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delta 687
zcmca|l<~?@#tmE8m^Zp@o_x@|aq<Fjo6RNc3pl|73%!0%77|i|OKwoenye%Y=5G!R
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diff --git a/test/IntegrationTests/refs/grhd_bondi_accretion/output1d.h5 b/test/IntegrationTests/refs/grhd_bondi_accretion/output1d.h5
index 740e709f24c311ee88cb745632d6f3fb0f3db056..0bafcfb8acc690abf8d822611a6f20431289162b 100644
GIT binary patch
delta 1180
zcmYk5Ur19?9LG6FOXl2e-08MLS)24C)yTIYF|mxa+Dys*ClpP68VrI6V$wrq;g;U=
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shDkK{a6#y<(1~t}iP+}ZZsBz&pPcQ?sHqB%nS?qn1iSb1IKJZj2aeyubpQYW

delta 1180
zcmYk6Ur19?9LG5)OLT5GuDUHymWy7b8trXJY^;>D+H52GpHSqVr@<hIASOLz7H-X3
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diff --git a/test/IntegrationTests/refs/grhd_bondi_infall_avmda/output1d.h5 b/test/IntegrationTests/refs/grhd_bondi_infall_avmda/output1d.h5
index 6f9227a415288f4c49e200f88648973114f66f4f..220135c6a0abbf3b14330cca8d8708b86cf1097e 100644
GIT binary patch
delta 31478
zcmYJ)cR1GJ`#*jam5{y3s6<Lx$vQ8)B+4!$BYP*wh}$f(S9YNy4YK!j+sVwRlo2vQ
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diff --git a/test/IntegrationTests/refs/srhd_simple_wave_avmda/output1d.h5 b/test/IntegrationTests/refs/srhd_simple_wave_avmda/output1d.h5
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diff --git a/test/UnitTests/src/test_spectral.jl b/test/UnitTests/src/test_spectral.jl
index 3a2a3860..cdb3f6f4 100644
--- a/test/UnitTests/src/test_spectral.jl
+++ b/test/UnitTests/src/test_spectral.jl
@@ -37,18 +37,14 @@ end
   f(x) = x^3 - 5.0 * x^2 + 1.0
   f2_ana = 118.0/21.0 # according to wolfram alpha
 
-  # use quadrature order N+1 = 3+1 = 4 here, because :LGL and :LGL_lumped are only exact up
-  # to and including 2N-1
+  # LGL quadrature is only exact up to order 2*3-1=5, so use N=4 instead
   N = 4
   L = 2.0
-  quadrs = [ :LGL, :GLGL, :LGL_lumped ]
-
-  for quadr in quadrs
-    el = SpectralElement(N, quadr)
-    fx = f.(el.z)
-    f2 = dg1d.inner_product(fx, fx, el)
-    @test isapprox(f2, f2_ana)
-  end
+  quadr = :LGL
+  el = SpectralElement(N, quadr)
+  fx = f.(el.z)
+  f2 = dg1d.inner_product(fx, fx, el)
+  @test isapprox(f2, f2_ana)
 
   # check that :GLGL quadrature is exact up to and including order 2N
   N = 3
@@ -64,8 +60,8 @@ end
 @testset "Barycentric interpolation" begin
 
   # setup
-  n_pts = 6
-  x, _ = dg1d.LGL.rule(n_pts)
+  Npts = 6
+  x, _ = dg1d.LGL.rule(Npts)
   bw = dg1d.barycentric_weights(x)
 
   # test function
@@ -83,7 +79,7 @@ end
   fxintrp = [ dg1d.barycentric_interp(xi, x, bw, fx) for xi in xsmpl ]
   @test all(isapprox.(fxintrp, fxsmpl))
 
-  # test interpolation using matrix multiplication 
+  # test interpolation using matrix multiplication
   I = dg1d.barycentric_interp_matrix(xsmpl, x)
   fxintrp = I * fx
   @test all(isapprox.(fxintrp, fxsmpl))
@@ -101,62 +97,83 @@ end
 end
 
 
-@testset "Operators GLGL" begin
+@testset "Test orthogonality relation GLGL" begin
 
-  # set up quadrature rule
-  N = 5
-  N_pts = N + 1
-  x, w, v = dg1d.GLGL.rule(N_pts)
-  D = dg1d.first_derivative_matrix(x)
+  Npts = 6
+
+  x, w, v, D = dg1d.GLGL.inner_product_rule(Npts)
+  L = zeros(Npts,Npts)
+  for i in 1:Npts
+    Pi = map(x->dg1d.legendre(x, i-1), x)
+    for j in i:Npts
+      Pj = map(x->dg1d.legendre(x, j-1), x)
+      lij = dg1d.GLGL.inner_product(Pi, Pj, x, w, v, D)
+      L[i,j] = lij
+      if i != j
+        L[j,i] = L[i,j]
+      end
+    end
+  end
+  dg1d.purge_zeros!(L)
 
   # analytical result
-  L_ana = zeros(N_pts, N_pts)
-  for i in 1:N_pts
+  L_ana = zeros(Npts, Npts)
+  for i in 1:Npts
     L_ana[i,i] = 2.0 / (2.0 * i - 1.0)
   end
 
-  L = dg1d.GLGL.linear_dependency_matrix(x, w, v, D)
-
   @test isapprox(L, L_ana)
 
 end
 
 
-@testset "Operators LGL" begin
-
-  # set up quadrature rule
-  n_pts = 6
-  x, w = dg1d.LGL.rule(n_pts)
-  quad(f1, f2) = dg1d.LGL.inner_product(w, f1, f2)
-
-  # analytical result
-  L_ana = zeros(n_pts, n_pts)
-  for i in 1:n_pts
-    L_ana[i,i] = 2.0 / (2.0 * i - 1.0)
+function lgl_linear_dependency_matrix_exact(Npts::Integer)
+  Npts2 = Npts+2
+  x2, w2 = dg1d.LGL.rule(Npts2)
+  L = zeros(N,N)
+  for i in 1:N
+    Pi = map(x->dg1d.legendre(x, i-1), x2)
+    for j in i:N
+      Pj = map(x->dg1d.legendre(x, j-1), x2)
+      lij = dg1d.LGL.integrate(Pi, Pj, x2, w2)
+      L[i,j] = lij
+      if i != j
+        L[j,i] = L[i,j]
+      end
+    end
   end
+  dg1d.purge_zeros!(L)
 
-  L = dg1d.LGL.linear_dependency_matrix_exact(x, w)
-
-  @test isapprox(L, L_ana)
-
+  return L
 end
 
 
 @testset "Test orthogonality relation" begin
 
-  # set up quadrature rule
-  n_pts = 6
-  x, w = dg1d.LGL.rule(n_pts)
-  quad(f1, f2) = dg1d.LGL.inner_product(w, f1, f2)
+  Npts = 6
+
+  Npts2 = Npts+2
+  x2, w2 = dg1d.LGL.rule(Npts2)
+  L = zeros(Npts,Npts)
+  for i in 1:Npts
+    Pi = map(x->dg1d.legendre(x, i-1), x2)
+    for j in i:Npts
+      Pj = map(x->dg1d.legendre(x, j-1), x2)
+      lij = dg1d.LGL.inner_product(Pi, Pj, x2, w2)
+      L[i,j] = lij
+      if i != j
+        L[j,i] = L[i,j]
+      end
+    end
+  end
+  dg1d.purge_zeros!(L)
 
   # analytical result
-  L_ana = zeros(n_pts, n_pts)
-  for i in 1:n_pts
+  L_ana = zeros(Npts, Npts)
+  for i in 1:Npts
     L_ana[i,i] = 2.0 / (2.0 * i - 1.0)
   end
 
-  L = dg1d.LGL.linear_dependency_matrix_exact(x, w)
-
   @test isapprox(L, L_ana)
 
 end
@@ -164,16 +181,15 @@ end
 
 @testset "More integrator tests" begin
 
-  n_pts = 6
+  Npts = 6
   p = x -> (x-4)^2 * (x+1)^3
 
-  x, w = dg1d.LGL.rule(n_pts)
+  x, w = dg1d.LGL.rule(Npts)
   int = dg1d.LGL.integrate(p, x, w)
   @test int ≈ 140/3
 
-  x, w, v = dg1d.GLGL.rule(n_pts)
-  D = dg1d.first_derivative_matrix(x)
-  int = dg1d.GLGL.integrate(p, x, w, v, D)
+  x, w = dg1d.GLGL.rule(Npts)
+  int = dg1d.GLGL.integrate(p, x, w)
   @test int ≈ 140/3
 
 end
-- 
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