Solve ID Burgers Equation for given Conditions Using MATLAB

Question

Consider the ID Burgers equation for the unknown u = u (x, t),

With initial condition 

                                     u(x, t) = 0.25 + 0.5sin( x) 

and periodic boundary conditions. Divide the computational domain (-1  into M equally spaced intervals and let the discrete solution u_i be defined at the center of each element x_i = -1 + (i- ½) 2/M. Add so-called ghost cells at i=0 and I = M+1 to the vector of unknowns and implement the periodic boundary conditions by setting

                                  u₀ = u_M      u_M+1 = u₁

• Determine u(x, t = 0.15) using a first order TVD scheme for M = 40. Plot the correct weak solution vs x together with the initial solution, and give the solution at t = 0.15 in a table together with x_i.
• Determine the correct weak solution for u(x, t = 20 using a first order scheme for M= 40. Add the solution to the plot and table of a).

Required:

• 1 Clearly annotated plot of u as a function of x at t = 0, 0.15, and 2 using a 1^st-order TVD method for M = 40;
• 1 table containing x and the solution u at t = 0, 0.15 and 2 using a first order TVD method for M = 40;

Summary

This question belongs to MATLAB software and discusses about application of MATLAB in mathematics to solve ID Burgers equation with initial and periodic boundary conditions and to determine first order TVD scheme and correct weak solution.

Answer is in MATLAB format