High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

Abstract

The main purpose of the paper is to show how to use implicit–explicit Runge–Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton’s iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R–K. We adopt several semi-implicit R–K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction–diffusion, convection diffusion and nonlinear diffusion system of equations.

Appendix

Here we prove that it is possible to apply the IMEX schemes to the general system (8) with no doubling of the stage fluxes. We start observing that by choosing $\hat{y}=(t,u)$ and $z=u$, we can rewrite system (1) as

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{d\hat{y}}{dt}(t) \,=\, \left( \begin{array}{c}1\\ \mathcal {H}(\hat{y}(t),z(t))\end{array}\right) , \\ \, \\ \displaystyle \frac{d z}{dt}(t) \,=\, \mathcal {H}(\hat{y}(t),z(t)), \end{array} \right. \end{aligned}$$

(37)

with initial conditions $\hat{y}(t_0) = (t_0,u_0)$, $z(t_0) = u_0$.

In this way, system (37) for $(\hat{y},z)$ is a particular case of an autonomous partitioned system in which $\mathcal {F}_1=(1, \mathcal {H})$ and $\mathcal {F}_2=\mathcal {H}$ but apparently with an additional computational cost since we double the number of variables. Now, we apply the partitioned Runge–Kutta method (6)–(7) to (37) obtaining

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\hat{k}}_i \,= \, \mathcal {H}\left( {\hat{Y}}_i,\, Z_i\right) , \quad 1\le i \le s, \\ \, \\ \displaystyle \ell _i \,=\, \mathcal {H}\left( {\hat{Y}}_i, \,Z_i\right) , \quad 1\le i \le s, \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\hat{Y}}_i \,=\, {\hat{y}}^n \,+\, \Delta t \,\sum _{j = 1}^{s}{\hat{a}}_{i,j}\,{\hat{k}}_j, \quad 1\le i \le s, \\ \displaystyle Z_i \,=\, z^n \,+\, \Delta t \,\sum _{j = 1}^{s}a_{ij}\,\ell _j, \quad 1\le i \le s, \end{array}\right. \end{aligned}$$

Using the assumption (5), that is, $\sum _j \hat{a}_{i,j}=\hat{c}_i$, we obtain that the first component of ${\hat{Y}}$ at the stage i is equal to $t^n+{\hat{c}}_i\Delta t$, and therefore, under the consistency condition $\sum _i {\hat{b}}_i=1$, the first component of ${\hat{y}}$ at time $t^{n+1}$ is equal to time $t^{n+1}$.

Replacing $\hat{y}$ by (t, y), we have

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle k_i \,= \, \mathcal {H}\left( t^n+{\hat{c}}_i\,\Delta t,\,Y_i,\, Z_i\right) , \quad 1\le i \le s, \\ \, \\ \displaystyle \ell _i \,=\, \mathcal {H}\left( t^n+\,{\hat{c}}_i\Delta t,\,Y_i, \,Z_i\right) , \quad 1\le i \le s, \end{array}\right. \end{aligned}$$

(38)

with

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle Y_i \,=\, y^n \,+\, \Delta t \,\sum _{j = 1}^{s}{\hat{a}}_{i,j}\,k_j, \quad 1\le i \le s, \\ \displaystyle Z_i \,=\, z^n \,+\, \Delta t \,\sum _{j = 1}^{s}a_{ij}\,\ell _j, \quad 1\le i \le s, \end{array}\right. \end{aligned}$$

Equation (38) already shows that $k_i=\ell _i, i=1,\ldots , s$. The numerical solutions at the next time step are

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle y^{n+1} \,=\, y^n \,+\, \Delta t \,\sum _{i = 1}^{s}{\hat{b}}_{i}\,k_i, \\ \displaystyle z^{n+1} \,=\, z^n \,+\, \Delta t \,\sum _{i = 1}^{s}b_{i}\,k_i. \end{array}\right. \end{aligned}$$

At this stage let us address several issues: number of evaluations, storage, order of accuracy and embedded methods.

Remark 5.1

Concerning the number of evaluations of $\mathcal {H}$, we observe that by writing $(\mathcal {G})$ as an autonomous partitioned system, we have $\ell _i=k_i$ for all $1\le i\le s$. Therefore only one evaluation of $\mathcal {H}$ is needed in (38), and only one set of stage fluxes is computed. Note that in (37), we could also choose $y=u$ and $\hat{z} = (t,u)$ and therefore use the $(c_i)_i$ coefficients for time stages.