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2.6 Stiffness matrices S

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

                                (               )
∫                          ∫      ∑
   F (Uˆ(ξ,η,t)) ⋅∇ φj(ξ,η ) =   F      Ui(t)φi(ξ,η)  ⋅∇ φj(ξ,η).
 T                          T      i

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 φi(ξ,η). This simplifies our stiffness term to

∫    ( ˆ      )                ∫  ∑
 T F  U (ξ,η,t)  ⋅∇ φj(ξ,η) ≈    T   iF(Ui(t))φi(ξ,η) ⋅∇φj (ξ,η)
                                           ∫                           (2.12)
                           =   ∑i F (Ui(t))   φi(ξ,η)⋅∇ φj(ξ,η).

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

∑    ( ∑              )
   F      Ui(t)φi(ξi,ηi)  φj(ξ,η) ≈ Sj ⋅G(Ui(t))+ Sj ⋅H (Ui(t))
 i      i                          x             y
with Sj selecting the j-th row of matrix S (see Appendix A.1.4).