# 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 $$\frac{\partial u}{\partial t} + u_0 \frac{\partial u}{\partial x} = \alpha \frac{\partial ^2 u}{\partial x^2} $$ ### Numerical Schemes ### Application ## Diffusion Equation $$\frac{\partial u}{\partial t} = \alpha \frac{\partial ^2 u}{\partial x^2} $$ The diffusion equation is parabolic. ### Numerical Schemes ### Application The diffusion equation is applied when a quantity $u$ is being distributed throughout a space or body with out moving the initial source of the quantity. #### 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. #### 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. ### Numerical Schemes ### Application ## Wave Equation $$\frac{\partial ^2 u}{\partial t^2} = \alpha_0 ^2 \frac{\partial ^2 u}{\partial x^2}$$ The linear version of the eqaution is $$\frac{\partial u}{\partial t} + \alpha_0 \frac{\partial u}{\partial x} = 0$$ ### Numerical Schemes ### Application # 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__.

### FTCS $$ \frac{u_j^{n+1} - u_j ^n}{\Delta t} = \alpha \frac{u_{j +1}^n -2 u_j^n + u_{j-1}^n}{\Delta x^2}$$

## FVM ## FEM ## Spectral method ## Spectral element methods # Termonology