The following table gives examples of basis functions based on Gauss-Lobatto points and their nodal points up to order 2.
Polynomial | Nodal point | |
Degree 0: | φ_{0}(x,y) := 1 | (,) |
Degree 1: | φ_{0}(x,y) := 1 - x - y | (,) |
φ_{1}(x,y) := x | (,0) | |
φ_{2}(x,y) := y | (0,) | |
Degree 2: | φ_{0}(x,y) := 1 - 3x + 2x^{2} - 3y + 4xy + 2y^{2} | (0,0) |
φ_{1}(x,y) := 4x - 4x^{2} - 4xy | (,0) | |
φ_{2}(x,y) := -x + 2x^{2} | (1,0) | |
φ_{3}(x,y) := 4y - 4xy - 4y^{2} | (0,) | |
φ_{4}(x,y) := 4xy | (,) | |
φ_{5}(x,y) := -y + 2y^{2} | (0,1) | |
The orthogonal Jacobi polynomials on triangle basis up to degree 2 are given in the following Table. Compared to the Gauss-Lobatto Points, the Jacobi polynomials are constructed hierarchically.
Polynomial | ||
Degree 0: | φ_{0}(x,y) := | |
Degree 1: | φ_{0}(x,y) := | |
φ_{1}(x,y) := -2 + 6y | ||
φ_{2}(x,y) := 2(2x - 1 + y) | ||
Degree 2: | φ_{0}(x,y) := | |
φ_{1}(x,y) := -2 + 6y | ||
φ_{2}(x,y) := (1 - 8y + 10y^{2}) | ||
φ_{3}(x,y) := 2(2x - 1 + y) | ||
φ_{4}(x,y) := 3(-1 + 5y)(2x - 1 + y) | ||
φ_{5}(x,y) := (1 - 2y + y^{2} - 6x + 6xy + 6x^{2}) | ||
For nodal basis functions of degree 1 and 2 with Gauss-Lobatto points, the inverse mass matrices are given by
Using orthonormal triangle basis functions based on normalized Jacobi Polynomials, the inverse mass matrix is identical to the identity matrix for arbitrary degree.
For Gauss-Lobatto nodal points, this leads to stiffness matrices each with a single zero-row for degree 1:
Computing the matrices x and y for degree 2, this still leads to almost dense matrices.
The matrices created for the orthonormal basis function have a higher sparsity pattern. Stiffness
matrices for degree 1 are given by
and for degree 2
Those matrices have clearly a sparser layout and are thus better suited for computation considering
the stiffness matrices only so far.
However, using modal basis functions, they have to be transferred to nodal basis functions. Using
Gauss-Lobatto nodal points and 1^{st} degree basis functions, this results in the following three
matrices:
and to the following matrices for 2^{nd} degree
Only a subset of conserved quantities is involved in the flux computation, see the following matrices for GL nodal points of order 2:
After flux computations, the flux updates only involve the nodal points of the edge for which the flux was computed for. Examples for such matrices generated for flux computations and Gauss Lobatto nodal points in the order of hypotenuse, right and left edge are given by
For RK with 2 stages, one possible tableau is
a_{1,1} := 0 | a_{1,2} := 0 |
a_{2,1} := | a_{2,2} := 0 |
b_{ 1} := 0 | b_{2} := 1 |
yielding
Another formulation for 2^{nd} order accuracy is given by Heun’s method, yielding the tableau
a_{1,1} := 0 | a_{1,2} := 0 |
a_{2,1} := 1 | a_{2,2} := 0 |
b_{1} := | b_{2} := |
The tableau for RK3 [But64] yielding 3-rd order accuracy is given by
a_{1,1} := 0 | a_{1,2} := 0 | a_{1,3} := 0 | |
a_{2,1} := | a_{2,2} := 0 | a_{2,3} := 0 | |
a_{3,1} := -1 | a_{3,2} := 2 | a_{3,3} := 0 | |
b_{1} := | b_{2} := | b_{3} := | |
The tableau for classical RK4 yielding 4-th order accuracy [SM03] is given by
a_{1,1} := 0 | a_{1,2} := 0 | a_{1,3} := 0 | a_{1,4} := 0 |
a_{2,1} := | a_{2,2} := 0 | a_{2,3} := 0 | a_{2,4} := 0 |
a_{3,1} := 0 | a_{3,2} := | a_{3,3} := 0 | a_{3,4} := 0 |
a_{4,1} := 0 | a_{4,2} := 0 | a_{4,3} := 1 | a_{4,4} := 0 |
b_{1} := | b_{2} := | b_{3} := | b_{4} := |
Here, we present the rotational invariancy of the Euler equations [Tor01]: For an edge normal _{e} pointing towards (cos(α),sin(α))^{T }, the rotation matrix R(α)_{e} is given by
We test for rotational invariancy by putting the flux terms from Eq. (1.9) into the rotational invariancy formula given in Eq. 2.9 which yields for the left hand side
| (A.2) |
and for the right hand side