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2.5 Mass matrix M

Discretizing the time step term from Eq. (2.9). We can factor out U by considering that φ are functions only depending on spatial parameters. This yields

∫  ∑                         ∑        ∫
      Ui(t)φi(ξ,η)φj(ξ,η ) =     Ui(t)   φi(ξ,η)φj(ξ,η).
 T  i  dt                     i  dt   ◟T-----◝◜------◞
                                     Mass matrix entry Mj,i

The integrals on the right-hand side of Eq. (2.10) can then be precomputed and their values stored to the mass matrix M. Using vector notation U⃗, we get a matrix-vector formulation

  ∑   Ui(t)                    ⃗
 T     dt  φi(ξ,η )φj (ξ,η) = M  ⋅Ut

with U⃗t also a matrix in case of several conserved quantities.

See Appendix A.1 for examples of matrices derived in this section and used in our simulations. Those matrices are computed with Maple1 , a computer algebra system for symbolic computations.

Applying the explicit Euler time-stepping method, this can then be rearranged to

U(t+ Δt ) = U(t)+ Δt ⋅M  -1(...),
thus requiring the computation of the inverse of the mass matrix (see Section A.1.3 for examples). Higher-order integration in time is further discussed in Sec. 2.14.