explicit

Definitions of explicit time integration schemes for conservative methods.

class CRK4(mesh, root=None, **default)

Class responsible for implementing an explicit 4th-order (classic) Runge-Kutta time-marching scheme that updates the current solution (\(t = t^{n}\)) to the next time step (\(t = t^{n+1}\)). Assuming a standard DG formulation,

\[\begin{split}\bm{k}_{1} &= -\widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{U}^{n} \big),\\[2ex] \bm{k}_{2} &= -\widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{U}^{n} + \bm{k}_1 / 2 \big),\\[2ex] \bm{k}_{3} &= -\widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{U}^{n} + \bm{k}_2 / 2 \big),\\[2ex] \bm{k}_{4} &= -\widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{U}^{n} + \bm{k}_3 \big),\\[2ex] \bm{U}^{n+1} &= \bm{U}^{n} + \big( \bm{k}_1 + 2\bm{k}_2 + 2\bm{k}_3 + \bm{k}_4 \big) / 6,\end{split}\]

where \(\widetilde{\bm{M}} = \bm{M} / \delta t\) is the weighted mass matrix, \(\bm{M}\) is the mass matrix and \(\bm{f}\) arises from the spatial discretization of the PDE.

class ExplicitEuler(mesh, root=None, **default)

Class responsible for implementing an explicit (forwards-)Euler time-marching scheme that updates the current solution (\(t = t^{n}\)) to the next time step (\(t = t^{n+1}\)). Assuming a standard DG formulation,

\[\widetilde{\bm{M}} \bm{U}^{n+1} = \widetilde{\bm{M}} \bm{U}^{n} - \bm{f} \big( \bm{U}^{n} \big),\]

where \(\widetilde{\bm{M}} = \bm{M} / \delta t\) is the weighted mass matrix, \(\bm{M}\) is the mass matrix and \(\bm{f}\) arises from the spatial discretization of the PDE.

class ExplicitSchemes(mesh, root=None, **default)

class SSPRK3(mesh, root=None, **default)

Class responsible for implementing an explicit 3rd-order strong-stability-preserving Runge-Kutta time-marching scheme that updates the current solution (\(t = t^{n}\)) to the next time step (\(t = t^{n+1}\)), see Section 4.1, Equation 4.2 in [1]. Assuming a standard DG formulation,

\[\begin{split}\bm{y}_{1} &= \bm{U}^{n} - \widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{U}^{n} \big),\\[2ex] \bm{y}_{2} &= \frac{3}{4} \bm{U}^{n} + \frac{1}{4} \bm{y}_{1} - \frac{1}{4} \widetilde{\bm{M}}^{-1} \bm{f} \big( \bm{y}_{1} \big),\\[2ex] \bm{U}^{n+1} &= \frac{1}{3} \bm{U}^{n} + \frac{2}{3} \bm{y}_{2} - \frac{2}{3} \widetilde{\bm{M}}^{-1 }\bm{f} \big( \bm{y}_{2} \big),\end{split}\]

where \(\widetilde{\bm{M}} = \bm{M} / \delta t\) is the weighted mass matrix, \(\bm{M}\) is the mass matrix and \(\bm{f}\) arises from the spatial discretization of the PDE.