4.8 Higher-order time stepping: Runge-Kutta

With DG simulations and their higher-order spatial discretization, the time-stepping method should be of a similar (higher-)order. To determine the framework requirements, we selected the explicit Runge-Kutta (RK) method. Considering the demands and algorithms shown in Section 2.14, RK methods require storing conserved quantities at particular points in time to V _i and their corresponding derivative D_i. Computing RK time step updates can then be achieved by additional stacks S^{V ₀} for V ₀ and S^D_i for D_i computed in each stage [BBSV10]. For an explicit RKn method assuring accuracy up to n-th order with V ₁ := V ₀ due to a_k,k = 0, we compute each stage i ∈{1,…,n} with the following algorithm:

Algorithm: RK time stepping

Before iterating over the RK stages, the cell data S_f^simCellData at the current time step is copied to S^{V ₀}.
For i in (1,…,n) do:

(a)

Compute D_i := R(V _i):
The simulation cell data stack S_f^simCellData is assumed to be set to V _i (see next step), the conserved quantities computed within the current RK stage. The time step-typical computations including edge communications are executed for S_f^simCellData. However, instead of updating the conserved quantities, only the change of the conserved quantities over time is stored to S_f^simCellData, yielding D_i.

(b)

Compute V _i := V ₀ + Δt∑ _j=1ⁿa_i,jD_j:
After the grid traversal, D_i is copied to S_f^D_i. Then V _i is computed and stored to S_f^simCellData by iterating over all elements of the stacks associated to V ₀ and D_j and applying equation (2.29).

Finally the time step is computed with

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Since we use pointers to mark the beginning of the stack for both push and pop operations, it is not necessary to copy stack data when assigning e.g. V ₀ := U. Instead of copying the entire stack, we can efficiently swap the stack pointers.