1.)3 point forward difference: Du/dx = aU(i) + bU(i+1)+cU(i+2)

2.)3 point backward difference: Du/dx = aU(i) + bU(i-1)+cU(i-2)

1)Explicit scheme with the first order two point forward differencing for (dU/dx).

2)Explicit scheme with first order two point backward differencing for (dU/dx).

1)Law-Wendroff One Step Scheme

2)MacCormack Two Step Scheme

3)Second-Order Upwind Scheme

4)4th Order Runge-Kutta Scheme

The wave equation is: du/dt + du/dx=0. The initial solution is u(x,0)=sin (2*n*pi*x/40), 0<=x<=40. The analytical solution will be u(x,t)=sin (2*n*pi*(x-t)/40). Periodic condition will be used at the two ends x=0 and x=40. Use 41 grid point mesh with Dx=1 and compute to t=18.

a)Solve the problem with n=1, 3 and CFL=1.0, 0.6, 0.3. Compare the solution and analytical solution and also compare the solutions between the results. Give your comments on the behavior of those results and the reason.

b)Compare the Maximum CFL number numerically allowed for each scheme that can be stable.

All the results should be plotted. Hand in your computer code with your results and your discussion.

You are encouraged to write your own code. However, you can also start with the attached FORTRAN code in case you are not familiar with programming.

1)Vertical line Gauss-Seidel iteration sweep from i-imax to i=i0.

2)Horizontal line Gauss-Seidel iteration sweep from j=j0 to j=jmax.

1)Generate uniform mesh in both x and y-direction.

2)Generate the mesh clustered near the wall as shown in the class notes by adjusting the stretching parameter, a.

Grid Generation Using Elliptic PDEs Generate the grid for a cylinder with the outer boundary located at 5 times of the radius by solving the Laplace's equations.

Requirements:

Two cases: 1) uniform mesh; 2) clustered near the wall.

1)Use Gauss-Seidel Point Iteration

2)Use Gauss-Seidel Line Iteration

3)Use Gauss-Seidel Line Iteration with SOR (optional, 10% bonus)

In your report, you need to present:

1)Introduction (why to use different methods above)

2)Governing Equations

3)Discretization Schemes

4)Boundary Condition Treatment

5)Results and Discussion, including description of the detailed geometry, compare the efficiency of different methods (convergency history), compare the mesh quality, etc.

1)Lax first order scheme

2)Lax-Wendroff One Step Scheme

3)MacCormack Two Step Scheme

4)4th Runge-Kutta Scheme

Use 51 mesh points in x-direction, repeat your solutions with CFL=1.0, 0.6, 0.3 and compare your solutions with the analytical solutions (given in your notes). The initial solution is u=1.0 for the first 11 mesh points, and u=0 for the rest of the mesh points.

All the results should be plotted. Hand in your computer code with your results and your discussion.

Following AIAA Paper 99-3348, solve 1D Euler equations for the 1D shock tube problem (Sod Problem) using Zha scheme and Roe scheme. Use control volume method, explicit Euler scheme, and 1st order accuracy in both spatial and temporal direction. Assume the pressure ration is 10:1. The initial pressure on the left side is 2.0E05 and on the right side is 2.0E04, the temperature is 303k on both sides, the shock tube length is 14m, the tube area iis 1m^2, initially, the diaphrame is located at x=7m. Compare your results with the analytical results attached at time level=0.01s. Use 201 uniform mesh points in space.

1)viscous shear stress terms tau_xx, tau_xz

2)inviscid flux F_bar and G_bar (Notes, p137)