Project: Steady Navier-Stokes Equations

The purpose of this project is to implement multigrid methods for solving steady state Navier-Stokes equations in two dimensions.

Problem Setting: Driven Cavity
Step 1: Multigrid of MAC for Stokes Equations
Step 2: FAS for Nonlinear Convection Diffusion Equation
Step 3: FAS for Navier-Stokes Equations with low Reynold Number
Step 4: Defect correction for High Reynold Number

Problem Setting: Driven Cavity

Driven cavity problem. The domain is [-1,1]^2. Navier-Stokes equation with zero source and zero Dirichlet boundary condition except on the top:

$\{ y=1; 0<= x <= 1 | u = 1, v = 0 \}$

Step 1: Multigrid of MAC for Stokes Equations

This is the Part III of Project: Fast Solvers for Stokes Equations. You can download my implementation: StokesMAC.zip Run squareStokesMAC.m to check the almost second order convergence. Read StokesVcycle.m to understand components of the algorithm.

Step 2: FAS for Nonlinear Convection Diffusion Equation

Implement FAS for solving the two-point boundary value problem

$$ - u''(x) + u(x)u'(x) = f(x), 0<x<1, u(0) = u(1) = 0. $$

The nonlinear Gauss-Seideal using the centeral difference scheme can be found in the book: A Multigrid Tutorial.

Step 3: FAS for Navier-Stokes Equations with low Reynold Number

Combine code from Step 1 and Step 2 to solve the Driven Cavity problem with low Reynold number or equivalently big visicosity constant.

Test your FAS for nu = 1 first and then try nu = 4h, 2h, h.

Step 4: Defect correction for High Reynold Number

Try your FAS for nu = 1e-6. If you want to keep working on this, please talk to me.