Understanding FVM(Lax Friedrich scheme) by solving Burger equation

Problem Description:

Project Description:

1. Burgers Equation: Burgers equation is a quadratic differential equation which is derived from the legendary Navier-Stokes equation.
?u/?t+?g(u)/?x=0
Where, g(u)= u^2/2
2. Finite Volume Method (FVM): The finite volume method is a method for representing and evaluating the partial differential equation in the form of algebraic equations. Here finite volume (cell) refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contains divergence terms are converted to surface integrals using Gauss divergence theorem. These terms are the evaluated as fluxes at the surfaces of each finite volume. These are conservative in nature since the amount of flux leaving a surface is equal to the amount of flux entering into the finite volume.
3. Lax Friedrich’s scheme: Several schemes are implemented on the basis of the finite volume method, one of which is Lax Friedrich’s scheme. To avoid the dependency of the solution on the direction of information flow, a central solver can be preferred. Lax-Friedrich’s scheme is one of the central solvers which can be used to solve a flow problem. Use Local Lax-Friedrich scheme for a better result.

Project Implementation:

1. At first, learn basic of numerical methods. Learn how FVM is different from other methods, what is a node, discretization, errors in numerical methods, truncation error, stability and convergence.
2. Then, learn what are the limitations in using FDM as a numerical approach to solve the differential equation.
3. Find the Numerical solution to that equation using Lax-Friedrich’s scheme. In this step, you have to write codes to discretize the domain, initial the boundary condition and interact the solution for the number of times until you get an accuracy of 10e-5 and achieve timestep 1s. Don’t forget to save the result data in a text file. Solve for both the direction of flow.
4. Then solve the same problem using Local Lax-Friedrich’s scheme.
5. Plot the result data using Gnuplot or Minitab.
6. Compare both the result and observe the difference.

Software requirements:

1. Dev-C++: You will be needing Dev-C++ software to write logic and interact the solution for a number of times.
2. Gnuplot: Also, you will be needing plotting software such as Gnuplot to plot the result data and compare the solution.

• Dev-C++
• Gnuplot

• Numerical methods
• solution to inviscid burger equation
• Lax Friedrichs scheme
• FVM