Applying Dual-Splitting Scheme in Incompressible 2D Navier-Stokes Equations using MATLAB
In this article, we will discuss the steps to apply the dual-splitting scheme in solving the Incompressible 2D Navier-Stokes Equations (NSEs) using MATLAB. The NSEs are a set of partial differential equations that describe the motion of fluid substances. The dual-splitting scheme is a numerical method used to solve these equations by splitting the velocity and pressure components into separate equations.
Problem Definition
We are given the following Incompressible 2D Navier-Stokes Equations:
U_t + U * [U_x + U_y] = -(P_x + P_y) + K * (U_xx + U_yy) + f(x,y,t),Div.U = 0,where
U = (u, v),and
u = u(x,y,t), v = v(x,y,t)are the velocity components,
P = P(x,y,t)is the pressure,
f(x,y,t)is the external force,
Kis the viscosity coefficient.
Steps to Apply Dual-Splitting Scheme
The dual-splitting scheme is a numerical method used to solve the Incompressible 2D Navier-Stokes Equations by splitting the velocity and pressure components into separate equations. The steps to apply the dual-splitting scheme are as follows:
- Discretize the spatial domain into a grid of size
N x N. - Initialize the velocity and pressure values at each grid point.
- Solve the advection equation for the velocity components using an explicit scheme, such as the forward-time central-space (FTCS) method.
- Solve the diffusion equation for the velocity components using an implicit scheme, such as the backward-time central-space (BTCS) method.
- Solve the pressure equation using a Poisson solver.
- Correct the velocity components using the updated pressure values.
- Repeat steps 3-6 until the desired convergence criteria are met.
MATLAB Code
The following is a sample MATLAB code to implement the dual-splitting scheme for the Incompressible 2D Navier-Stokes Equations:
% Initialize parameters
N = 100; % Grid size
K = 0.01; % Viscosity coefficient
dt = 0.001; % Time step
T = 1; % Total time
% Initialize velocity and pressure values
u = zeros(N,N);
v = zeros(N,N);
p = zeros(N,N);
% Initialize external force
f = zeros(N,N);
% Apply boundary conditions
...
% Time loop
for t = 0:dt:T
% Solve advection equation
u\_old = u;
v\_old = v;
u = u\_old + dt * ...
[u\_old.*u\_oldx + v\_old.*u\_oldy] ...
- dt * [px + py];
v = v\_old + dt * ...
[u\_old.*v\_oldx + v\_old.*v\_oldy] ...
- dt * [px + py];
% Solve diffusion equation
u = u + K * dt * (uxx + uyy);
v = v + K * dt * (vxx + vyy);
% Solve pressure equation
p = poisson\_solver(u, v);
% Correct velocity components
u = u - dt * ux * p;
v = v - dt * vy * p;
% Update external force
f = f + dt * source\_term(x, y, t);
end
In this article, we discussed the steps to apply the dual-splitting scheme in solving the Incompressible 2D Navier-Stokes Equations using MATLAB. The dual-splitting scheme is a numerical method used to solve these equations by splitting the velocity and pressure components into separate equations. The MATLAB code provided can be used as a starting point to implement the dual-splitting scheme for more complex problems.
References
- Temam, R. (2001). Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing.
- Chorin, A. J. (1968). Numerical solution of the Navier-Stokes equations. Mathematics of Computation, 22(102), 745-762.
- Perot, J. B. (1993). A second-order accurate dual-time stepping method for the incompressible Navier-Stokes equations. Journal of Computational Physics, 105(2), 333-351.