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Effective velocity, or phase velocity, for analytical and numerical time integration respectively:

$$v_{ef}= \left| \frac{\lambda_I}{k} \right|=\left| \frac{\phi}{k\Delta t} \right|\quad;\quad\phi=\arctan \left( \frac{G_I}{G_R} \right)$$

Contents

• Explicit Euler
• Implicit Euler
• Crank-Nicolson
• Rugen-Kutta Schemes

Explicit Euler

$$\begin{gathered} \mathbf u^{n+1}=\mathbf u^n+\Delta t\mathbf F(\mathbf u^n,t_n)\\ \left. \frac{\dd u}{\dd t} \right|_{t_n}=F(u^n,t_n)=F^n \end{gathered}$$

The gain is, assuming $F=\lambda u$:

$$G=\frac{u^{u+1}}{u^n}=\frac{u^n+\Delta t\lambda u^n}{u^n}=1+\Delta t\lambda$$

• 1st order scheme

Implicit Euler

$$\mathbf u^{n+1}=\mathbf u^n+\Delta t\mathbf F(\mathbf u^{n+1},t_{n+1})$$

The gain is:

$$u^{n+1}-\Delta\lambda u^{n+1}=u^n\quad \rightarrow \quad G=\frac{u^{n+1}}{u^n}=\frac{1}{1-\Delta t\lambda}$$

• 1st order scheme

Crank-Nicolson

Time discretization with $t_{n+1/2}$:

$$\mathbf u^{n+1}=\mathbf u^n+\frac{\Delta t}{2}(\mathbf F(\mathbf u^n,t_n)+\mathbf F(\mathbf u^{n+1},t_{n+1}))$$

The gain is:

$$G=\frac{1+\Delta t\lambda/2}{1-\Delta t\lambda/2}$$

• 2nd order scheme

Rugen-Kutta Schemes

Multiphase scheme. Schemes from 2 to 4 phases:

$$\begin{aligned} \mathbf u^{n+1}&=\mathbf u^n+\frac{1}{2}(\mathbf k_1+\mathbf k_2)\\ \mathbf u^{n+1}&=\mathbf u^n+\frac{1}{6}(\mathbf k_1+4\mathbf k_2+\mathbf k_3)\\ \mathbf u^{n+1}&=\mathbf u^n+\frac{1}{6}(\mathbf k_1+2\mathbf k_2+2\mathbf k_3+\mathbf k_4) \end{aligned}$$

The gain depends on the number of phases:

$$G=1+\Delta t\lambda+\frac{1}{2!}(\Delta t\lambda)^2+\frac{1}{3!}(\Delta t\lambda)^3+\frac{1}{4!}(\Delta t\lambda)^4$$