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As computers cannot work with an infinite domain, discretization of the domain is required.

Contents

• Spatial Discretization
  • Finite Differences
  • Finite Volumes
• Temporal Discretization
• Space-Time Discretization

Spatial Discretization

There are different methods of spatial discretization:

• Finite differences: Based on differential equations
• Finite volumes: Based on integral equations
• Finite elements
• Spectral

Given the analytical solution in a time $t$ and defined in all values of $x$:

$$u=u(x,t)$$

Spatial discretization:

• Nodes: $x_i=(i-1)\Delta x$
  • $i=1,2,\dots,N$
  • $\Delta x=L/(N-1)$
  • Alternatively, $x_i=i\Delta x$ with $i=0,1,2,\dots,N+1$ where $\Delta x=L/(N+1)$
• Values: $u_i(t)=u(x_i,t)$
  • $i=1,2,\dots,N$
  • For simplicity, $u_i=u_i(t)$

Nodes $i\in\{1,N\}$ are boundary nodes, where values of $u_i$ should be imposed.

Finite Differences

• Based on differentials
• Domain represented by a distribution of points (nodes)
• Derivatives are approximated with the linear combination of the values of variables being derived
• Consistency condition: The weight of each terms of the lineal combinations must be chosen such that they converge to the corresponding derivative when the nodes are infinitesimally close

Given a generic node $i$, its value is defined as $x_i=(i-1)\Delta x$ where $\Delta x=L/(N-1)$

• Internal nodes: $x_i\rightarrow i\in[1,N-1]$
  • Computational variables: $u_i\rightarrow i\in[1,N-1]$ for each time step
• External (boundary) nodes: $x_0=0,x_N=L$
  • Boundary conditions: $u_0,u_N$

Example of a 2nd order extrapolation: $\tilde u_N=2u_{N-1}-u_{N-2}$:

$$\begin{aligned} \tilde u_N=2u_{N-1}-u_{N-2}\approx&2\left( u_N-\partial_xu_n \Delta x+\frac{1}{2}\partial_{xx}u_n\Delta x^2+\cdots \right)\\ &-\left( u_N-2\partial_xu_n\Delta x+\frac{1}{2}\partial_{xx}u_n(2\Delta x)^2+\cdots \right)\\ \approx&u_N-O(\Delta x^2) \end{aligned}$$

• Consistency: $\lim_{\Delta_x\rightarrow0}\tilde u_N=u_N$
• For simplicity, $u_N=\tilde u_N$

2nd order central derivative:

$$\begin{gathered} u_{i+1}=u_i+\Delta xu'_i+\frac{(\Delta x)^2}{2!}u_i''+\frac{(\Delta x)^3}{3!}u_i'''+\frac{(\Delta x)^4}{4!}u_i'''+\cdots\\ u_{i-1}=u_i-\Delta xu'_i+\frac{(\Delta x)^2}{2!}u_i''-\frac{(\Delta x)^3}{3!}u_i'''+\frac{(\Delta x)^4}{4!}u_i'''-\cdots\\ \end{gathered}$$

Summing both expressions, we can only retain 2nd derivative:

$$\begin{gathered} u_{i+1}+u_{i-1}=2u_i+(\Delta x)^2u_i''+\frac{2}{4!}(\Delta x)^4u_i''''+\cdots\\ u_i''=\frac{u_{i+1}-2u_i+u_{i-1}}{(\Delta x)^2}-\frac{1}{12}(\Delta x)^2u_i''''+\cdots \end{gathered}$$

The dominant term scales with $\Delta x^2$, therefore we can define the second derivatives as:

$$\mathcal D^2u_i=\frac{u_{i+1}-2u_i+u_{i-1}}{(\Delta x)^2}$$

• Consistency: $\lim_{\Delta x\rightarrow0}\mathcal D^2u_i=u_i''$

A list of delayed, advanced, and centered discretizations:

$$\begin{aligned} &\mathcal Du_i\approx\frac{u_i-u_{i-1}}{\Delta x}\quad&;&\quad\mathcal D^2u_i\approx\frac{u_i-2u_{i-1}+u_{i-2}}{\Delta x^2}\\ &\mathcal Du_i\approx\frac{u_{i+1}-u_1}{\Delta x}\quad&;&\quad\mathcal D^2u_i\approx\frac{u_{i+2}-2u_{i+1}+u_i}{\Delta x^2}\\ &\mathcal Du_i\approx\frac{u_{i+1}-u_{i-1}}{2\Delta x}\quad&;&\quad\mathcal D^2u_i\approx\frac{u_{i+1}-2u_i+u_{i-1}}{\Delta x^2}\\ \end{aligned}$$

Finite Volumes

Both NS and Euler equations can be written in the conservative form:

$$\frac{\partial\mathbf U}{\partial t}+\nabla\cdot\mathbf F=\mathbf S$$

• $\mathbf U$: Vector of conservative variables
• $\mathbf F$: Vector of fluxes. Sum of convective $\mathbf F^c$ and viscous $\mathbf F^v$ fluxes
  • $\mathbf F^v$ is also the sum of physical viscosity and numerical viscosity

Integrating the equation in a control volume $p$ gives:

$$V_p\frac{\dd\overline{\mathbf U}_p}{\dd t}+\int_{V_p}\nabla\cdot\mathbf F\dd\mathbf V=\int_{V_p}\mathbf S\dd\mathbf V$$

• $\overline{\mathbf U}_p=\frac{1}{V_p}\int_{\partial V_p}\mathbf U\dd\mathbf V$: Mean of the computational variable in the control volume $p$,

Using Gauss:

$$V_p\frac{\dd\overline{\mathbf U}_p}{\dd t}+\oint_{\partial V_p}\mathbf F\cdot\dd\mathbf S=\int_{V_p}\mathbf S\dd\mathbf V$$

Given the control volumes are plain faces, this can be simplified to:

$$\frac{\dd\overline{\mathbf U}_p}{\dd t}+\frac{1}{V_p}\sum_{f\in\partial V_p}^\text{faces}\mathbf F_f\cdot\Delta\mathbf A_f=\overline{\mathbf S}$$

Temporal Discretization

Now considering the solution of any node $x_i$, in function of the time: $u_i(t)$. The time discretization is defined as:

$$t_n=n\Delta t\quad;\quad n=0,1,2,\dots\quad;\quad\Delta t>0$$

• $t_{n+1}=(n+1)\Delta t=t_n+\Delta t$
• The value of $u_i$ in time $t_n$ can be represented as $u_i^n$.

Space-Time Discretization

Using the method of lines:

• Spatial discretization: $\partial_tu_i+c\partial_xu_i=0$
• Temporal discretization: $u_i^{n+1}=u_i^n+\Delta tF_i^n$
  • $F_i^n=-u_i\partial_xu_i$ at $t_n$