Parallel-in-Time High-Order Multiderivative IMEX Solvers

Functional‐preserving predictor‐corrector multiderivative schemes

“…[13], see also ref. [12], applied to the entropy‐conserving nonlinear oscillator with $$T_{end}= 100$$. Left are numerical results for $$\Delta t= 0.5$$, right are results for $$\Delta t= 0.2$$.…”

“…In this work, we rely on three Runge‐Kutta schemes, two with two‐derivatives, see ref. [12, Equations (2) and (3), respectively, for the Butcher tableaux], and a two‐point three‐derivative scheme with Butcher tableau $$$$\begin{align*} c = \def\eqcellsep{&}\begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad B^{(1)} = \def\eqcellsep{&}\begin{pmatrix} 0 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}, \quad B^{(2)} = \def\eqcellsep{&}\begin{pmatrix} 0 & 0 \\ \frac{1}{10} &- \frac{1}{10} \end{pmatrix}, \quad B^{(3)} = \def\eqcellsep{&}\begin{pmatrix} 0 & 0 \\ \frac{1}{120} & \frac{1}{120} \end{pmatrix}. \end{align*}$$$$The final HBPC scheme to be presented here relies on a predictor ($$k=0$$) and correction steps ($$1 \le k \le k_{\max }$$) for the quantities $$w^{n,l}$$.…”

“…It is hence a slight generalization of ref. [11, 12], where only schemes with $$c_s = 1$$ are treated. In any case, the update step is explicit.…”

“…[11] and then subsequently extended to higher orders in refs. [12,13]. HBPC has shown favorable behavior for the solution of compressible flow equations [9,14,15].…”

Jacobian-free explicit multiderivative Runge-Kutta methods for hyperbolic conservation laws

“…and so on. Using this approach, one can derive stable, high-order and storage-efficient schemes very easily [28]. This can be extended to partial differential equations (PDEs) with a time-component, such as Eq.…”