# Model Equations ## Burger's Equation $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial ^2 u}{\partial x^2} $$ The inviscid version of the equation is $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0$$ The linear version of the Burger's Equation is often called the Convection–Diffusion equation $$\frac{\partial u}{\partial t} + u_0 \frac{\partial u}{\partial x} = \alpha \frac{\partial ^2 u}{\partial x^2} $$ ### Discretization Schemes ### Application ## Diffusion Equation $$\frac{\partial u}{\partial t} = \alpha \frac{\partial ^2 u}{\partial x^2} $$ The diffusion equation is parabolic. ### Discretization Schemes ### Application #### Heat Conduction $$\frac{\partial T}{\partial t} = \alpha \frac{\partial ^2 T}{\partial x^2} $$ where $T$ is the temperature, and $alpha$ is the heat conduction constant. In 2D the equation will be #### Flow in porous media $$\frac{\partial u}{\partial t} = c \frac{\partial ^2 u}{\partial x^2} $$ ## Poisson Equation $$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = f(x, y)$$ Setting $f(x, y) = 0 $ will give the Laplace equation. The equation is elliptic. The equation can be used to express 2D steady heat conduction : $$\frac{\partial }{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial }{\partial y}\left(k\frac{\partial T}{\partial y}\right) = 0$$ ### Discretization Schemes #### Central Difference ##### 2D Steady Heat Equation The 2D steady heat conduction equation can be dicretized by finite differences to $$k\frac{T_{i+1, j} - 2T_{i, j} + T_{i-1, j}}{(\Delta x)^2} + k\frac{T_{i+1, j} - 2T_{i, j} + T_{i-1, j}}{(\Delta y)^2} = 0$$ $$\left(\frac{2k}{(\Delta x)^2 + (\Delta y)^2}\right) T_{i, j} = \frac{k}{(\Delta x)^2}T_{i-1, j} + \frac{k}{(\Delta x)^2}T_{i+1, j} + \frac{k}{(\Delta y)^2}T_{i, j-1} + \frac{k}{(\Delta y)^2}T_{i, j+1} $$ which can be written on a more compact notation $$a_P T_{i, j} = a_W T_{i-1, j} + a_E T_{i+1, j} + a_S T_{i, j-1} + a_N T_{i, j+1}$$ where $a_W = a_E = \frac{k}{(\Delta x)^2}$, $ a_S = a_N = \frac{k}{(\Delta x)^2}$, and $a_P = a_W + a_E + a_S + a_N$ At boundaries with __Dirichlet boundary condition__ (i.e. boundaries with a constant value that does not change), the cell which is "in" the boundary is not included in the computational domain. Instead, the neighbouring cell is solved with a special equation. If the boundary is immidiatly below cell $(x_i, y_1)$, then the value $T_{i, 1}$ in the cell is calculated by $$ a_P T_{i, 1} = a_W T_{i -1, 1} + a_E T_{i+1, 1} + a_N T_{i, 2} + \tilde{S}_u$$

## Wave Equation $$\frac{\partial ^2 u}{\partial t^2} = \alpha_0 ^2 \frac{\partial ^2 u}{\partial x^2}$$ The Wave Equation is hyperbolic. ## Linear Advection Equation $$\frac{\partial u}{\partial t} + \alpha_0 \frac{\partial u}{\partial x} = 0$$ The Linear Advection Eqaution is hyperbolic. The exact solution is $$ u = u_0 f(x - \alpha_0 t)$$ ### Discretization Schemes #### FTCS The FTCS scheme of the advection equation is _unconditionally unstable_. $$u^{n+1}_j = u^n_j - \frac{\alpha_0 \Delta t}{\Delta x}(u^n_{j+1} - u^n_{j-1}) $$ #### Explicit Upwind Scheme $$u^{n+1}_j = u^n_j - \frac{\alpha_ 0 \Delta t}{\Delta x} (u^n_j - u^n_{j-1}) $$ The scheme is stable in the interval $$0 \space \leq \space \frac{\alpha_ 0 \Delta t}{\Delta x} \space \leq \space 1$$ $$ TE = \mathcal{O}(\Delta t, \Delta x)$$ #### Implicit Upwind Scheme $$u^{n+1}_j = \frac{u^n_j + \frac{\alpha_0 \Delta t}{\Delta x} u^{n+1}_{j-1}}{1 + \frac{\alpha_0 \Delta t}{\Delta x}} $$ # Boundary Conditions # Numerical Methods ## Euler $$ y_{n+1} = y_n + h \cdot f(t_n, y_n)$$ ### Errors $$ e_{n+1} = \mathcal{O} (h^2)$$ $$ E_n = \mathcal{O} (h)$$ ### Stability Region $$ \{z \in \mathbb{C} |\quad |1 + z| \leq 1\}$$ ## Implicitt Euler $$ y_{n+1} = y_n + h \cdot f(t_{n+1}, y_{n+1})$$ ### Errors $$ E_n = \mathcal{O} (h)$$ ### Stability Region $$ \{z \in \mathbb{C} |\quad |\frac{1}{1- z}| \leq 1\}$$ ## Trapezoidal $$y_{n+1} = y_n + h\cdot \frac{1}{2}\cdot [f(x_n, y_n) +(f(x_{n+1}, y_{n+1})] $$ ### Errors $$ e_{n+1} = \mathcal{O} (h^3)$$ $$ E_n = \mathcal{O} (h^2)$$ ### Stability region $$ \{z \in \mathbb{C} | Re(z) < 0\}$$ ## Heun $$y_{n+1} = y_n + h\cdot \frac{1}{2}\cdot [f(x_n, y_n) +(f(x_{n+1}, y*_{n+1})] $$ where $y*_{n+1} = y_n + h\cdot f(x_n, y_n) $ ### Errors $$ e_{n+1} = \mathcal{O} (h^3)$$ $$ E_n = \mathcal{O} (h^2)$$ ### Stability Region $$ \{z \in \mathbb{C} |\quad |1 + z + \frac{z^2}{2}| \leq 1\}$$ ## Runge-Kutta $$ k_1 = f(t_n, y_n)$$ $$ k_2 = f(t_n + \frac{h}{2}, y_n + \frac{h}{2}\cdot k_1)$$ $$ k_3 = f(t_n + \frac{h}{2}, y_n + \frac{h}{2}\cdot k_2)$$ $$ k_2 = f(t_n + h, y_n + h\cdot k_3)$$ $$ y_{n+1} = y_n + \frac{h}{6} \cdot (k_1 + 2k_2 + 2k_3 + k4)$$ ### Errors $$ e_{n+1} = \mathcal{O} (h^5)$$ $$ E_n = \mathcal{O} (h^4)$$ ### Stability Region $$ \{z \in \mathbb{C} |\quad |1 + z + \frac{z^2}{2}+ \frac{z^3}{6}+ \frac{z^4}{24}| \leq 1\}$$ # Discretization methods ## FDM ## Definitions for FDM #### General Finite Difference Methods approximates in-between states and derivatives as $$ \frac{d u(x_{j +\frac{1}{2}})}{dx} \approx \frac{u_{j +1} - u_j}{\Delta x}$$ $$ \frac{d^2 u(x_j)}{dx^2} \approx \frac{u_{j +1} -2 u_j + u_{j-1}}{\Delta x^2}$$ #### Consistency A FDM of a PDE is called __consistent__ if it approximates the PDE such that the truncation error TE = FDM - PDE goes to zero as $\Delta x $ and $\Delta t$ goes to zero. #### Stability A FDM of a PDE is called __stable__, if the 2-norm of the FDM solution $ \mathbf{u}^n $ stays bounded for any time level $n$. Translated to english; for an initial value problem with inital condition $u(x, 0) = u^0(x)$, $ \mathbf{u}^n $ satisfies $$\parallel\mathbf{u}^n \parallel _2 \space\leq \space Ke^{\alpha t_n}\parallel\mathbf{u}^0 \parallel _2$$ where $K$ and $\alpha$ are constants independent $\mathbf{u}^0$, $\Delta x$, and $\Delta t$. Usually $ \alpha = 0$. Stability for a discretization method is derived through __von Neumann stability analysis__, also known as Fourier analysis. This analysis requires the following 1. The FDM is linear. 2. The FDM has constants coefficients. 3. The problem has __periodic__ boundary conditions. 4. The grid is __equidistant__. #### Convergence and Lax Equivalance Theorem If a consistent FDM is stable, it will also be convergent. The order of convergence will be equal the order of accuracy. ### FTCS $$ \frac{u_j^{n+1} - u_j ^n}{\Delta t} = \alpha \frac{u_{j +1}^n -2 u_j^n + u_{j-1}^n}{\Delta x^2}$$ #### Accuracy $$ TE = \mathcal{O}(\Delta t) + \mathcal{O}(\Delta x^2)$$ Order of convergence is equal to order of accuracy $$\parallel\mathbf{u}^n - u(\mathbf{x}, t_n) \parallel _2 = \mathcal{O}(\Delta t) + \mathcal{O}(\Delta x^2)$$ #### Stability AN FTCS scheme is stable when $$ 0 \leq \frac{\alpha \Delta t}{\Delta x^2} \leq \frac{1}{2} $$ ### BTCS - simple implcit $$ \frac{u_j^{n+1} - u_j ^n}{\Delta t} = \alpha \frac{u_{j +1}^{n+1} -2 u_j^{n+1} + u_{j-1}^{n+1}}{\Delta x^2}$$ #### Accuracy $$ TE = \mathcal{O}(\Delta t) + \mathcal{O}(\Delta x^2)$$ #### Stability BTCS is __unconditionally stable__. $$ |g(k \Delta x)| = \frac{1}{1 + 4\frac{\alpha \Delta t}{\Delta x^2} sin^2(\frac{k \Delta x}{2})} \leq 1$$ BTCS is convergent for all $$\frac{\alpha \Delta t}{\Delta x^2} \geq 0 $$ ### Explicit Upwind scheme ## FVM ## FEM ## Spectral method ## Spectral element methods # Termonology