Lagrange Derived Type

The number of target points.

The number of control points.

The boundary interpolation matrix that is used to map a grid of nodal values at the control points to the element boundaries.

The barycentric weights that are calculated from the controlPoints and used for interpolation.

The set of nodes in one dimension where data is known. To create higher dimension interpolation and differentiation operators, structured grids in two and three dimensions are created by tensor products of the controlPoints. This design decision implies that all spectral element methods supported by the Lagrange class have the same polynomial degree in each computational/spatial dimension. In practice, the controlPoints are the Legendre-Gauss, Legendre-Gauss-Lobatto, Legendre-Gauss-Radau, Chebyshev-Gauss, Chebyshev-Gauss-Lobatto, or Chebyshev-Gauss-Radau quadrature points over the domain [-1,1] (computational space). The Init routine for this class restricts controlPoints to one of these quadrature types or uniform points on [-1,1].

The derivative matrix for mapping function nodal values to a nodal values of the derivative estimate. The dMatrix is based on a strong form of the derivative.

The split-form derivative matrix D_split = D - 0.5M^{-1}B, where B is the SBP boundary operator.

D_split is skew-symmetric under the M inner product: MD_split + D_split^TM = 0. Unlike D, it is NOT an SBP operator (the weighted symmetric part is zero, not the boundary term B). This property makes D_split ideal for the EC-DGSEM split-form volume integral:

du/dt = -(2/J) sum_n D_split[n,i] * F_EC(u_i, u_n) + (1/J) M^{-1} B^T f_Riemann

The skew-symmetry of the volume term guarantees it contributes zero to the entropy rate, so all entropy change passes through the surface term. An entropy-dissipative Riemann solver then makes the full scheme entropy-stable.

Using D_split in the volume combined with plain f_Riemann on the surface is algebraically identical to using D in the volume with the penalty (f_Riemann - f_local) on the surface (Trixi.jl convention).

In SELF index convention (dSplitMatrix(ii,i) = D_split[i-1, ii-1]): dSplitMatrix(ii,i) = dMatrix(ii,i) - 0.5(bMatrix(i,2)bMatrix(ii,2) - bMatrix(i,1)*bMatrix(ii,1)) / qWeights(i)

The derivative matrix for mapping function nodal values to a nodal values of the derivative estimate. The dgMatrix is based on a weak form of the derivative. It must be used with bMatrix to account for boundary contributions in the weak form.

The interpolation matrix (transpose) for mapping data from the control grid to the target grid.

Mortar L2-projection matrices, the adjoints of mortarR under the exact L2 inner product on [-1,1]: given traces g_k(s) on the two sub-edges, the projected big-side trace is u(i) = sum_k sum_ii mortarP(ii,i,k)*g_k(ii).

The matrices are built as P_k = M^{-1} B_k with the exact (dense) 1-D mass matrix M(m,i) = int l_m l_i dxi and B_k(m,j) = (1/2) int l_m(xi_k(s)) l_j(s) ds, evaluated with an internal Gauss rule that is exact for the degree 2N integrands. Consequently sum_k matmul(P_k,R_k) = I (a polynomial split across both sub-edges is recovered) sum_m w_m (P_k g)_m = (1/2) sum_j w_j g_k(j) (discrete conservation) for any control node type. mortarP carries the 1/2 sub-edge Jacobian appropriate for projecting solution traces; when projecting surface-flux integrands (which carry the small side's nScale = nScale_big/2), scale by 2 (see MortarFluxCollect in the 2-D mapped data classes).

Mortar restriction matrices for 2:1 nonconforming (mortar) interfaces.

An element edge trace u(xi) = sum_i u_i l_i(xi) on [-1,1] ("big" side) is restricted to the two half-edges ("small" sides) of its 2:1 neighbors. Sub-edge k=1 occupies xi in [-1,0] and k=2 occupies xi in [0,1], with the sub-edge coordinate s in [-1,1] related to the big-edge coordinate through xi_1(s) = (s-1)/2 and xi_2(s) = (s+1)/2.

mortarR(ii,i,k) = l_ii( xi_k(s_i) ), where s_i are the control points; following the SELF matrix convention, the restricted trace is u_k(i) = sum_ii mortarR(ii,i,k)*u(ii). Restriction of a degree N polynomial is exact.

The quadrature weights for discrete integration. The quadradture weights depend on the type of controlPoints provided; one of Legendre-Gauss, Legendre-Gauss-Lobatto, Legendre-Gauss-Radau, Chebyshev-Gauss, Chebyshev-Gauss-Lobatto, Chebyshev-Gauss Radau, or Uniform. If Uniform, the quadrature weights are constant $$dx = \frac{2.0}{N+1}$$ .

The set of nodes in one dimension where data is to be interpolated to. To create higher dimension interpolation and differentiation operators, structured grids in two and three dimensions are created by tensor products of the targetPoints. In practice, the targetPoints are set to a uniformly distributed set of points between [-1,1] (computational space) to allow for interpolation from unevenly spaced quadrature points to a plotting grid.