Jacobian-Free Explicit Multiderivative Runge–Kutta Methods for Hyperbolic Conservation Laws

Abstract

Based on the recent development of Jacobian-free Lax–Wendroff (LW) approaches for solving hyperbolic conservation laws (Zorio et al. in J Sci Comput 71:246–273, 2017, Carrillo and Parés in J Sci Comput 80:1832–1866, 2019), a novel collection of explicit Jacobian-free multistage multiderivative solvers for hyperbolic conservation laws is presented in this work. In contrast to Taylor time-integration methods, multiderivative Runge–Kutta (MDRK) techniques achieve higher-order of consistency not only through the excessive addition of higher temporal derivatives, but also through the addition of Runge–Kutta-type stages. This adds more flexibility to the time integration in such a way that more stable and more efficient schemes could be identified. The novel method permits the practical application of MDRK schemes. In their original form, they are difficult to utilize as higher-order flux derivatives have to be computed analytically. Here we overcome this by adopting a Jacobian-free approximation of those derivatives. In this paper, we analyze the novel method with respect to order of consistency and stability. We show that the linear CFL number varies significantly with the number of derivatives used. Results are verified numerically on several representative testcases.

A Butcher Tableaux

In this section, we show the multiderivative Runge–Kutta methods used in this work through their Butcher tableaux. We use three two-derivative methods taken from [5], see Tables 3, 4 and 5; two three-derivative methods taken from [21], see Tables 6 and 7; and one four-derivative method, constructed for this paper, see Table 8. This last scheme has been derived from the idea that it should be of form

for , with update

These forms have also been used in [5] and [21].

Table 3 $2\text {DRK}3\text {-}2$:Third order two-derivative Runge–Kutta scheme using two stages [5]

Table 4 $2\text {DRK}4\text {-}2$:Fourth order two-derivative Runge–Kutta scheme using two stages [5]

Table 5 $2\text {DRK}5\text {-}3$:Fifth order two-derivative Runge–Kutta scheme using three stages [5]

Table 6 $3\text {DRK}5\text {-}2$:Fifth order three-derivative Runge–Kutta scheme using two stages [21]

Table 7 $3\text {DRK}7\text {-}3$:Seventh order three-derivative Runge–Kutta scheme using three stages [21]. The coefficients are given by $c_2 = \frac{3 - \sqrt{2}}{7}$, $c_3 = \frac{3 + \sqrt{2}}{7}$, $a_{32}^{(3)} = \frac{122 + 71\sqrt{2}}{7203}$, $b_1^{(3)} = \frac{1}{30}$, $b_2^{(3)} = \frac{1}{15} + \frac{13\sqrt{2}}{480}$, $b_3^{(3)} = \frac{1}{15} - \frac{13\sqrt{2}}{480}$

Table 8 $4\text {DRK}6\text {-}2$:Sixth order four-derivative Runge–Kutta scheme using two stages