# TEP4280: Introduction to Computational Fluid Dynamics

# Model Equations

## Burger's Equation

The Burger's equation is parabolic.

The inviscid version of the equation is

The linear version of the Burger's Equation is often called the Convection–Diffusion equation

### Discretization Schemes

### Application

## Diffusion Equation

### Discretization Schemes

### Application

#### Heat Conduction

In 2D the equation will be

#### Flow in porous media

## Poisson Equation

Setting

The equation is elliptic.

The equation can be used to express 2D steady heat conduction :

### Discretization Schemes

#### Central Difference

##### 2D Steady Heat Equation

The 2D steady heat conduction equation can be dicretized by finite differences to

which can be written on a more compact notation

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

For **Neumann boundary condition**, a similar approach is to be taken. The core equation describing

## Wave Equation

The Wave Equation is hyperbolic.

## Linear Advection Equation

The Linear Advection Eqaution is hyperbolic.

The exact solution is

### Discretization Schemes

#### FTCS

The FTCS scheme of the advection equation is *unconditionally unstable*.

#### Explicit Upwind Scheme

#### Implicit Upwind Scheme

# Boundary Conditions

# Numerical Methods

## Euler

### Errors

### Stability Region

## Implicitt Euler

### Errors

### Stability Region

## Trapezoidal

### Errors

### Stability region

## Heun

### Errors

### Stability Region

## Runge-Kutta

### Errors

### Stability Region

## Solution of Linear Systems of Equations

Most systems will be on the form

### Direct Methods

Direct methods will provide the exact solution for the gien equation.

For problems where **tridiagonal matrix**, it can be solve dwith direct methods. A tridiagonal matrix will only have non-zero entries in its three central diagonals, resulting in the following system

Implicit diffusion equation, implicit linear advection equation and similar are appropriate to solve with direct methods

*other* entries on the same row

and for at least one row must be *strictly* larger than the sum of the other entris in the same row

For tridiagonal matricies, that only have the elements

#### TDMA

If the tridiagonal matrix *diagonally dominant*, the system can be solved with **TDMA**, which is a simplified form of Gaussian elimination. TDMA can solve the system in

The process is split up in 4 distinct steps

##### Step 1: LU Decomposition

We assume that the matrix

where

The original system of equations

#### Cholesky Factorization

If

### Iterative Methods

Iterative methods requires and initial guess to start its procedures.

# Discretization methods

## FDM

## Definitions for FDM

#### General

Finite Difference Methods approximates in-between states and derivatives as

#### 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

#### Stability

A FDM of a PDE is called **stable**, if the 2-norm of the FDM solution

Stability for a discretization method is derived through **von Neumann stability analysis**, also known as Fourier analysis.
This analysis requires the following

- The FDM is linear.
- The FDM has constants coefficients.
- The problem has
**periodic**boundary conditions. - 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

#### Accuracy

Order of convergence is equal to order of accuracy

#### Stability

AN FTCS scheme is stable when

### BTCS - simple implcit

#### Accuracy

#### Stability

BTCS is **unconditionally stable**.

BTCS is convergent for all