TEP4280: Introduction to Computational Fluid Dynamics
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
In 2D the equation will be
Flow in porous media
The equation is elliptic.
The equation can be used to express 2D steady heat conduction :
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
The Wave Equation is hyperbolic.
Linear Advection Equation
The Linear Advection Eqaution is hyperbolic.
The exact solution is
The FTCS scheme of the advection equation is unconditionally unstable.
Explicit Upwind Scheme
Implicit Upwind Scheme
Solution of Linear Systems of Equations
Most systems will be on the form
Direct methods will provide the exact solution for the gien equation.
For problems where
Implicit diffusion equation, implicit linear advection equation and similar are appropriate to solve with direct methods
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
If the tridiagonal matrix
The process is split up in 4 distinct steps
Step 1: LU Decomposition
We assume that the matrix
The original system of equations
Iterative methods requires and initial guess to start its procedures.
Definitions for FDM
Finite Difference Methods approximates in-between states and derivatives as
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
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.
Order of convergence is equal to order of accuracy
AN FTCS scheme is stable when
BTCS - simple implcit
BTCS is unconditionally stable.
BTCS is convergent for all