# Optimal Methods

• For nonlinear/nonautonomous systems:
• Explicit methods
• Implicit methods
• For linear, autonomous systems:
• Explicit methods
• Implicit methods

# Maple Scripts

• First Order

Optimal first order explicit SSP Runge-Kutta methods consist simply of repeated forward Euler steps. These methods all have effective SSP coefficient equal to unity and are equivalent to simply using the forward Euler method.

• Second Order
• The $m$ stage, second order SSP methods: $$\begin{eqnarray} u^{(0)} &=& u^n \\ u^{(i)} &=& \left( 1+ \frac{\Delta t}{m-1} L \right) u^{(i-1)} , i=1 ,..., m-1 \\ u^{m} & = & \frac{1}{m} u^{(0)} + \frac{m-1}{m} \left( 1+ \frac{\Delta t}{m-1} L \right) u^{(m-1)} \\ u^{n+1} & = & u^{(m)} \end{eqnarray}$$ have an optimal SSP coefficient $c= m-1$ among all methods with nonnegative coefficients. Although these methods were designed for linear problems, they are also nonlinearly second order Each such method uses $m$ stages to attain the order usually obtained by a $2$-stage method, but has SSP coefficient $c=m-1$, thus the effective SSP coefficient here is $c_{\text{eff}} = \frac{m-1}{m}$.

• $m$-Stage, Order $m$
• The class of $m$ stage schemes given by: $$\begin{eqnarray} u^{(i)} & = & u^{(i-1)} + \Delta t L u^{(i-1)}, \qquad i=1, . . . , m-1 \\ u^{(m)} & = & \sum_{k=0}^{m-2} \alpha_{m,k} u^{(k)} + \alpha_{m,m-1} \left(u^{(m-1)}+\Delta t L u^{(m-1)} \right), \end{eqnarray}$$ where $\alpha_{1,0} = 1$ and $$\begin{eqnarray} \alpha_{m,k} &=& \frac{1}{k} \alpha_{m-1,k-1}, \qquad k=1, . . . , m-2 \\ \alpha_{m,m-1} & = & \frac{1}{m!}, \qquad \alpha_{m,0} = 1- \sum_{k=1}^{m-1} \alpha_{m,k} \end{eqnarray}$$ is an $m$-order linear Runge-Kutta method which is SSP with SSP coefficient $c=1$, which is optimal among all $m$ stage, $p=m$ order SSPRK methods with nonnegative coefficients. The effective SSP coefficient is $c_{\text{eff}} = \frac{1}{m}$.

• $m$-Stage, Order $m-1$
• The $m$ stage, order $p=m-1$ method: $$\begin{eqnarray} u^{(0)} &=& u^n \\ u^{(i)} & = & u^{(i-1)} + \frac{1}{2} \Delta t L u^{(i-1)}, \qquad i=1, . . . , m-1 \\ u^{(m)} & = & \sum_{k=0}^{m-2} \alpha_{m,k} u^{(k)} + \alpha_{m,m-1} \left( u^{(m-1)}+ \frac{1}{2} \Delta t L u^{(m-1)} \right), \\ u^{n+1} & = & u^{(m)} \end{eqnarray}$$ Where the coefficients are given by: $$\begin{eqnarray} \alpha_{2,0} & = & 0 \qquad \alpha_{2,1}=1 \\ \alpha_{m,k} &=& \frac{2}{k} \alpha_{m-1,k-1}, \qquad k=1, . . . , m-2 \\ \alpha_{m,m-1} & = & \frac{2}{m} \alpha_{m-1,m-2} , \qquad \alpha_{m,0} = 1- \sum_{k=1}^{m-1} \alpha_{m,k} \end{eqnarray}$$ is SSP with optimal (for methods with nonnegative coefficients) SSP coefficient $c=2$. The effective SSP coefficient for these methods is $c_{\text{eff}} = \frac{2}{m}$.