Differential Equations and Eigen Values
- Consider the linear constant coefficient hyperbolic system.
Write the Maxwell’s system of equations in matrix form. Determine the eigenvalues and eigenvectors of the normal flux matrix An=! ! “! + ! ” “” and compute the upwind flux for such a system.
- Consider a quadrilateral element [−1,1]^2 with vertices (r1,s1)=(−1,−1),(r2,s2)=(1,−1),(r3,s3)=(1,1),(r4,s4)=(−1,1).
- 1) Provide explicit formulas for degree 1 Lagrange polynomials on the reference quadrilateral, and use them to construct a mapped physical element with vertices (xi,yi) for i = 1,…, 4.
- 2) Explain how to compute geometric change of variables factors (e.g., ∂r/∂x,∂s/∂x, etc) for quadrilateral elements. Are they constant as they were for a triangular element? Give a geometric interpretation of when the geometric factors are constant for a quadrilateral.
- 3) Explain what would need to change in the implementation of the code to accommodate geometric factors which are non-constant.
- Consider a DG discretization of the Laplacian on a periodic uniform mesh using the BR1 formulation without penalty.
- 1) Explicitly characterize the null-space of the DG discretization matrix for degree N=0 polynomials. Hint: use the weak formulation.
- 2) Construct the the DG discretization matrix explicitly for a degree N=0 DG approximation with 8 elements.
