Differential Equations and Eigen Values
2. 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.
3. 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.
4. 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.
