Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

Abstract

Perturbed Runge–Kutta methods (also referred to as downwind Runge–Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge–Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.

Appendix

In this section we give additional details on SSP coefficients and optimal perturbations of second order 2-stage Runge–Kutta methods and the classical 4-stage fourth order Runge–Kutta method.

1.1 Second Order 2-Stage Methods

We consider the family of 2-stage second order methods (26). In Example 1 we studied perturbations that increase the SSP coefficient for the linear case. For nonlinear problems, in Example 3, Fig.  2 shows the values of $R^{\mathrm{opt}}(K)$ for $\alpha \in [-3,3]$.

In this section, for each $\alpha$, we give the expressions for $R^{\mathrm{opt}}(K)$ and we show optimal perturbations $\widetilde{K}_{NL}$ such that $R(K, \widetilde{K}_{NL})=R^{\mathrm{opt}}(K)$. It is important to point out the convenience of choosing $\widetilde{K}_{NL}= \widetilde{K}_{L}$, where $\widetilde{K}_{L}$ denotes the optimal perturbation for the linear case. In this case, we have not only $R(K, \widetilde{K}_{L})=R^{\mathrm{opt}}(K)$ but also ${R}_{\mathrm{Lin}}(K, \widetilde{K}_{NL})=R_{\mathrm{Lin}}^{\mathrm{opt}}(K)$. The computations required to obtain the results in this section have been done with the symbolic computation program Mathematica.

If we denote by $r=R^{\mathrm{opt}}(K)$, we have that

$$\begin{aligned} r= {\left\{ \begin{array}{ll} \displaystyle \frac{1}{|\alpha |}, &{} \qquad \displaystyle \hbox {if}\; \quad \alpha \in \left( -\infty , -\frac{1}{2} \left( 1+\sqrt{7}\right) \right] \bigcup \left[ \frac{1}{2} \left( -1+\sqrt{7}\right) , \infty \right) , \\ \displaystyle \frac{-1+\alpha +\sqrt{ 3 \alpha ^2-2 \alpha +1} }{|\alpha | }, &{} \qquad \displaystyle \hbox {if}\; \quad \alpha \in \left( -\frac{1}{2} \left( 1+\sqrt{7}\right) , 0\right) \bigcup \left( 0, \frac{1}{2} \left( -1+\sqrt{7}\right) \right) . \end{array}\right. }\nonumber \\ \end{aligned}$$

(72)

Next we give optimal perturbations $\widetilde{K}_{NL}$.

For $\alpha <0$, we obtain that it is not possible to obtain a perturbation of the form (26) with ${\tilde{b}}_2=0$ and ${\tilde{a}}_{21}=0$. Consequently, $\widetilde{K}_{NL}\ne \widetilde{K}_{L}$ and we always have that ${R}_{\mathrm{Lin}}(K, \widetilde{K}_{NL})<R_{\mathrm{Lin}}^{\mathrm{opt}}(K)$. Optimal perturbations of the form (26) for different values of $\alpha <0$ must satisfy the following conditions.

• For $-\frac{1}{2} \left( 1+\sqrt{7}\right) \le \alpha <0$, the coefficients ${\tilde{a}}_{21}, {\tilde{b}}_1$ and ${\tilde{b}}_2$ in $\widetilde{K}_{NL}$ must satisfy

  $$\begin{aligned} -\,\alpha \le {\tilde{a}}_{21}\le \frac{1-r \, \alpha }{2 \, r}, \qquad {\tilde{b}}_1=-\frac{r\, {\tilde{a}}_{21}}{2\, \alpha } , \qquad {\tilde{b}}_2=-\frac{1}{2\, \alpha }, \end{aligned}$$

  where $r=R^{\mathrm{opt}}(K)$.

• For $\alpha \le -\frac{1}{2} \left( 1+\sqrt{7}\right)$, we should have

  $$\begin{aligned} {\tilde{a}}_{21}=-\,\alpha ,\qquad -\frac{1}{2 \,\alpha }\le {\tilde{b}}_1\le \frac{-2 \,\alpha ^2-2 \, \alpha +1}{4\, \alpha }, \qquad -\,\frac{1}{2 \,\alpha }\le {\tilde{b}}_2\le \frac{2\, \alpha \,{\tilde{b}}_1-1}{4\, \alpha }. \end{aligned}$$

For $\alpha >0$ we can find optimal perturbations with ${\tilde{b}}_2=0$ and ${\tilde{a}}_{21}=0$. Coefficient ${\tilde{b}}_1$ must satisfy the following conditions.

• For $0<\alpha \le \left( -\,1+\sqrt{7}\right) /2$, we have that

  $$\begin{aligned} \qquad {\tilde{b}}_1=\frac{\sqrt{3 \alpha ^2-2 \alpha +1}-\alpha }{2 \alpha } . \end{aligned}$$

  (73)

  Thus there is a unique $\widetilde{K}_{NL}$ of the form (26). In this case, we have $R(K)<R(K, \widetilde{K}_{NL})=R^{\mathrm{opt}}(K)$.

• For $\left( -1+\sqrt{7}\right) /2<\alpha < 1$, we also get $R(K)<R^{\mathrm{opt}}(K)$, but in this case the optimal perturbation $\widetilde{K}_{NL}$ is not unique. All the perturbations with ${\tilde{b}}_1$ satisfying

  $$\begin{aligned} \frac{1-\alpha }{\alpha } \le {\tilde{b}}_1\le \frac{2 \alpha ^2-2 \alpha +1}{4 \alpha } , \end{aligned}$$

  are optimal. In particular, we can take $\widetilde{K}_{NL}=\widetilde{K}_{L}$. With this choice, $R(K, \widetilde{K}_{L})=R^{\mathrm{opt}}(K)=1/\alpha$ and ${R}_{\mathrm{Lin}}(K, \widetilde{K}_{L})=R_{\mathrm{Lin}}^{\mathrm{opt}}(K)\approx 1.22$. Furthermore, $\alpha =\left( -\,1+\sqrt{7}\right) /2$ provides the largest SSP coefficient within the family of 2-stage second order method (see Fig. 2).

• For $1\le \alpha$, we have $R(K)=R^{\mathrm{opt}}(K)=1/\alpha$ and the optimal perturbation $\widetilde{K}_{NL}$ is not unique. All the values

  $$\begin{aligned}0\le {\tilde{b}}_1\le \frac{2 \alpha ^2-2 \alpha +1}{4 \alpha } \end{aligned}$$

  give optimal perturbations. We can take $\widetilde{K}_{NL}=0$, but in this case ${R}_{\mathrm{Lin}}(K, 0)< R_{\mathrm{Lin}}^{\mathrm{opt}}(K)$. A better choice is $\widetilde{K}_{NL}=\widetilde{K}_{L}$. Observe that, for $\alpha =1$, we get the optimal SSP coefficient $R(K)=1$ that cannot be increased by perturbations.

Next, we consider some concrete values of $\alpha$ to show the the expressions of the perturbations. For each value, we give the Butcher tableau of the perturbation and matrices $\alpha ^{\mathrm{up}}$ and $\alpha ^{\mathrm{down}}$ in (36).

• For $\alpha =1/2$ we get method RK2a in [18] with $R(K)=0$. With perturbation

  $$\begin{aligned} \widetilde{K}&= \begin{pmatrix} 0 &{}\quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad 0\\ {\tilde{b}}_1 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \quad \alpha ^{\mathrm{up}}= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ {\tilde{b}}_1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 2 {\tilde{b}}_1 &{}\quad 0 \end{pmatrix}, \quad \alpha ^{\mathrm{down}}= \begin{pmatrix} 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0\\ 1-2{\tilde{b}}_1 &{}\quad 0 &{}\quad 0 \end{pmatrix},\nonumber \\ \gamma&= \begin{pmatrix} 1\\ 1-{\tilde{b}}_1\\ 0 \end{pmatrix}, \end{aligned}$$

  (74)

  where ${\tilde{b}}_1=\frac{1}{2} \left( \sqrt{3}-1\right)$, we get $R(K,\widetilde{K}) ={R}_{\mathrm{Lin}}(K, \widetilde{K})= \sqrt{3}-1$.

• For $\alpha =2/3$, we have a nontrivial SSP coefficient $R^{\mathrm{opt}}(K)=1/2$, but we can increase this value to $R(K,\widetilde{K}_1)=R^{\mathrm{opt}}(K)=1$ with perturbation

  $$\begin{aligned} \widetilde{K}_1&= \begin{pmatrix} 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0\\ 1/4 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \quad \alpha ^{\mathrm{up}}= \begin{pmatrix} 0 &{} \quad 0 &{}\quad 0 \\ 2/3 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 3/4 &{}\quad 0 \end{pmatrix} , \quad \alpha ^{\mathrm{down}}= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0\\ 1/4 &{}\quad 0 &{}\quad 0 \end{pmatrix} ,\\ \gamma&= \begin{pmatrix} 1\\ 1/3\\ 0 \end{pmatrix}. \end{aligned}$$

  For this perturbation, $R(\phi _K)={R}_{\mathrm{Lin}}(K, \widetilde{K}_1)=1$. We can take $\gamma =(1,0,0)^t$ by modifying the first column of $\alpha ^{\mathrm{up}}$ and $\alpha ^{\mathrm{down}}$ according to (50),

  $$\begin{aligned} \widetilde{K}_2&= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 1/6 &{}\quad 0 &{}\quad 0\\ 3/8 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \quad \alpha ^{\mathrm{up}}= \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ 5/6 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 3/4 &{} \quad 0 \end{pmatrix} , \quad \alpha ^{\mathrm{down}}= \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ 1/6 &{} \quad 0 &{} \quad 0\\ 1/4 &{} \quad 0 &{} \quad 0 \end{pmatrix} ,\\ \gamma&= \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}. \end{aligned}$$

• As it has been pointed out above, the largest value in the $\alpha$-family of 2-stage second order schemes is $R^{\mathrm{opt}}(K)=(1+\sqrt{7})/3$ and it is obtained for $\alpha =(\sqrt{7}-1)/2$. The perturbation is of the form (26) with ${\tilde{b}}_1= \left( \sqrt{7}-2\right) /2$, and

  $$\begin{aligned} \alpha ^{\mathrm{up}}= \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ 1 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad \frac{1}{9} \left( 4+\sqrt{7}\right) &{}\quad 0 \end{pmatrix} , \quad \alpha ^{\mathrm{down}}= \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0\\ \frac{1}{9} \left( 5-\sqrt{7}\right) &{} \quad 0 &{} \quad 0 \end{pmatrix} , \quad \gamma _r = \begin{pmatrix} 1\\ 0 \\ 0 \end{pmatrix} \end{aligned}$$

  This is the perturbation obtained in [9, Table V] by numerical search in the class of perturbations considered in [9].

1.2 Classical Fourth Order 4-Stage Method

For nonlinear problems, applying the analysis above, we find that the optimal perturbation of the classical method has SSP coefficient given by the real root of $x^3+2 x^2+4 x-4=0$, which is approximately $R^{\mathrm{opt}}(K)\approx 0.685016$. The corresponding perturbation is not unique. For instance, we can take $\gamma _r=(1,0,0,0,0)$, and all entries of $\alpha ^{\mathrm{down}}_r$ equal to zero except

$$\begin{aligned} \left( \alpha ^{\mathrm{down}}_r\right) _{31}&= \frac{r^2}{4}&\left( \alpha ^{\mathrm{down}}_r\right) _{42}&= \frac{r^2}{2}, \end{aligned}$$

(75)

where $r=R^{\mathrm{opt}}(K)$. However, there exist other optimal perturbations with additionally $(\alpha ^{\mathrm{down}}_r)_{42}=\epsilon$ where $0\le \epsilon \le 0.782$.

We remark that nearly-optimal perturbations for this method are given in [30, p. 448] and [15]. Interestingly, these different perturbed methods have different values of ${R}_{\mathrm{Lin}}(K, \widetilde{K})$.