Multiple-Relaxation Runge Kutta Methods for Conservative Dynamical Systems

Abstract

We generalize the idea of relaxation time stepping methods in order to preserve multiple nonlinear conserved quantities of a dynamical system by projecting along directions defined by multiple time stepping algorithms. Similar to the directional projection method of Calvo et. al. we use embedded Runge–Kutta methods to facilitate this in a computationally efficient manner. Proof of the accuracy of the modified RK methods and the existence of valid relaxation parameters are given, under some restrictions. Among other examples, we apply this technique to Implicit–Explicit Runge–Kutta time integration for the Korteweg–de Vries equation and investigate the feasibility and effect of conserving multiple invariants for multi-soliton solutions.

Appendices

A: List of RK Methods

See Tables 2, 3, 4, 5, 6 and 7.

Table 2 SSPRK(2,2): Second-order method $(A,\vec {b}^{\,1})$ with a first-order embedded method $(A,\vec {b}^{\,2})$

Table 3 SSPRK(3,3): Third-order method $(A,\vec {b}^{\,1})$ with $(A,\vec {b}^{\,2})$ and $(A,\vec {b}^{\,3})$ as second-order embedded methods

Table 4 Heun(3,3): Third-order method $(A,\vec {b}^{\,1})$ with a second-order embedded method $(A,\vec {b}^{\,2})$

Table 5 RK(4,4): Fourth-order method $(A,\vec {b}^{\,1})$ with a second-order embedded method $(A,\vec {b}^{\,2})$

Table 6 Fehlberg(6,4): Fifth-order method $(A,\vec {b}^{\,1})$ with a fourth-order embedded method $(A,\vec {b}^{\,2})$, and $(A,\vec {b}^{\,3})$ and $(A,\vec {b}^{\,4})$ as third-order embedded methods

Table 7 DP(7,5): Fifth-order method $(A,\vec {b}^{\,1})$ with $(A,\vec {b}^{\,2})$ and $(A,\vec {b}^{\,3})$ as fourth-order embedded methods

B: Soliton Solutions

Different soliton solutions [20] of the KdV equation (32) are given below:

• (a) 1-soliton solution:

  $$\begin{aligned} u(x,t)= \beta _1 {{\,\textrm{sech}\,}}^2(\xi _1) \;, \end{aligned}$$

  (37)

  where $\beta _1 = 1$ and $\xi _1 = \frac{\sqrt{\beta _1}(x-2 \beta _1 t)}{\sqrt{2}}$.

• (b) 2-soliton solution:

  $$\begin{aligned} u(x, t) = -\frac{2(\beta _1-\beta _2)\left( \beta _2 {{\,\textrm{csch}\,}}^2(\xi _2)+\beta _1 {{\,\textrm{sech}\,}}^2(\xi _1)\right) }{\left( \sqrt{2 \beta _1} \tanh (\xi _1)-\sqrt{2 \beta _2} \coth (\xi _2)\right) ^2} \;, \end{aligned}$$

  (38)

  where $\beta _1 = 0.5$, $\beta _2 = 1$, $\xi _1 = \frac{\sqrt{\beta _1}(x-2 \beta _1 t)}{\sqrt{2}}$, and $\xi _2 = \frac{\sqrt{\beta _2}(x-2 \beta _2 t)}{\sqrt{2}}$.

• (c) 3-soliton solution:

  $$\begin{aligned} u(x, t)= & {} \beta _1{{\,\textrm{sech}\,}}^2(\xi _1)- \nonumber \\{} & {} \frac{2(\beta _2-\beta _3)\left( \frac{2(\beta _3-\beta _1)\left( \beta _3 {{\,\textrm{csch}\,}}^2(\xi _3)-\beta _1 {{\,\textrm{sech}\,}}^2(\xi _1)\right) }{\left( \sqrt{2 \beta _3} \tanh (\xi _3)-\sqrt{2 \beta _1} \tanh (\xi _1)\right) ^2}-\frac{2(\beta _1-\beta _2)\left( \beta _2 {{\,\textrm{csch}\,}}^2(\xi _2)+\beta _1 {{\,\textrm{sech}\,}}^2(\xi _1)\right) }{\left( \sqrt{2 \beta _1} \tanh (\xi _1)-\sqrt{2 \beta _2} \coth (\xi _2)\right) ^2} \right) }{\left( \frac{2(\beta _1-\beta _2)}{\sqrt{2 \beta _1} \tanh (\xi _1)-\sqrt{2 \beta _2} \coth (\xi _2)}-\frac{2(\beta _3-\beta _1)}{\sqrt{2 \beta _3} \tanh (\xi _3)-\sqrt{2 \beta _1} \coth (\xi _1)}\right) ^2}, \nonumber \\ \end{aligned}$$

  (39)

  where $\beta _1 = 0.4$, $\beta _2 = 0.7$, $\beta _3 = 1$, $\xi _1 = \frac{\sqrt{\beta _1}(x-2 \beta _1 t)}{\sqrt{2}}$, $\xi _2 = \frac{\sqrt{\beta _2}(x-2 \beta _2 t)}{\sqrt{2}}$, and $\xi _3 = \frac{\sqrt{\beta _3}(x-2 \beta _3 t)}{\sqrt{2}}$.