Numerical Calculations (MAC Method)

The fundamental equations are,

(1)

(2)

Now, I need to brake them down to calculate them numerically.

The basic plan is below.

 

1. Make a divergence of the equation(2) so that we can substitute the equation(1) into the equation(2).

2. Make a equation of by substituting equation(1) into the equation(2).

3. Substitute current into the equation of to get new .

4. Substitute current and new into the equation(2) to get new .

Repeat step 3 and 4.

 

First, make the divergence of the equation(2). I drop the acceleration a to make it simple.

Rewrite it with and make it a equation of .

Approximate the time derivative by forward difference,

Here, the should be zero since the continuity equation, but it can't be exactly zero because of its error, so we leave for the error and treat only the next step as zero. So the equation will be,

Drop the first part of the right side because it's negligible,

To expanse this equation, try only part first, 

   

   

   

   

       

           

   

       

           

Here, ignore the derivative of cause it's small,

   

Put it back into the equation,

(a)

Put current into this equation to get .


Then put the new and current into the equation (2) to get new .

Approximate the time derivative of the equation (2) by forward difference,

Rewrite it as a equation for ,

(b)

Calculate next with this equation. Then put it into the equation (a) to get new , and so on.


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