These html pages are based on the

PhD thesis "Cluster-Based Parallelization of Simulations on Dynamically Adaptive Grids and Dynamic Resource Management" by Martin Schreiber.

There is also

more information and a PDF version available.

### 2.6 Stiffness matrices

For a partial evaluation of the stiffness terms, we follow the direct approach of expanding the
approximated solution, yielding

Since an accurate integration for flux functions with rational terms would be computationally
infeasible, approximations are typically used for this evaluation [Coc98,HW08,Sch03]. Such an
approximation is the nodal-wise evaluation of the flux term at nodal points and reconstruction of a
continuous function with the basis functions φ_{i}(ξ,η). This simplifies our stiffness term to

For the modal basis function, a projection to/from nodal basis is required. We continue by
replacing F with two flux functions G and H (see Section 1.1). Then, we can again evaluate integrals
and rearrange the equations to a matrix-matrix formulation

with
^{j} selecting the j-th row of matrix S (see Appendix A.1.4).