Help with Undergraduate Math, Algebra & PDE Implicit Scheming

[kriss2]

Hi
a want to code implicite scheme for PDEs
u_t =a laplace u + L p_t
p_t = laplace p + u

pls help
I made a classical FEM discretization, but i don't know how to programe it.

(u⁽ⁿ⁺¹⁾_ij - uⁿ)/t = a ((u⁽ⁿ⁺¹⁾_(i-1j) -2 u⁽ⁿ⁺¹)_(ij) +u⁽ⁿ⁺¹⁾_{(i+1 j)})
/x² + (u⁽ⁿ⁺¹)_{(i j-1)}-2u⁽ⁿ⁺¹⁾_(ij) + u^{(n+1)( i j+1 )}/y²) +L (p⁽ⁿ⁺¹⁾
-pⁿ)/t

(p⁽ⁿ⁺¹⁾_ij - pⁿ)/t = ((p⁽ⁿ⁺¹⁾_(i-1j) -2 p⁽ⁿ⁺¹⁾_(ij) +p⁽ⁿ⁺¹⁾_{(i+1 j)})
/x² + p⁽ⁿ⁺¹⁾_{(i j-1)}-2p⁽ⁿ⁺¹⁾_(ij) + p⁽ⁿ⁺¹⁾_{( i j+1 )}/y²) + uⁿ

 

[pmsrw3]

Well, it's not simple. What kind of help are you looking for? "i don't know how to programe it" is pretty broad. Do you want to know how to program a computer? Or are you asking how to efficiently solve a banded system of linear equations? Or what?

 

[kriss2]

I have explicite method and I need something similar for implicite metod

k=1:T
Pold=P;
Uold=U;
i=2:X-1;
j=2:Y-1;
P(i,j)=Pold(i,j) ...
+ ((Pold(i-1,j)-2*Pold(i,j)+Pold(i+1,j))/X^2 ...
+ (Pold(i,j-1)-2*Pold(i,j)+Pold(i,j+1))/Y^2)*tau ...
+ Uold(i,j)


U(i,j)=Uold(i,j)+ ...
a *( (Uold(i-1,j)-2*Uold(i,j)+Uold(i+1,j))/X^2 ...
+ (Uold(i,j-1)-2*Uold(i,j)+Uold(i,j+1))/Y^2) ...
+ L*(P(i,j) - Pold(i,j));


figure(1)
Pcolor(U)
caxis([0 1.2])
colorbar

end

 

[pmsrw3]

OK. For every i,j, you have an equation relating u_ij and p_ij at time n+1 to their values at time n. Let's suppose you have an m x m grid -- then these, together with the boundary conditions, give you 2m^2 equations in 2m^2 unknowns. You need to solve that big system of linear equations.

There are a few different algorithms for solving systems of linear equations. Good programming practice forbids re-inventing the wheel, and at this point you should find a library available for your software environment and use it. In particular, you want algorithms and libraries for "sparse" or "banded" matrices. Systems like this give very big matrices (e.g., if you're working on a 30 x 30 grid, you'll have an 1800 x 1800 matrix), most of whose elements are 0. A sparse matrix library will represent this matrix compactly (i.e., without using up 3240000 reals in your memory) and will solve it efficiently. Solving a general system of linear equations is O(n^3), but algorithms designed for sparse systems will do it in O(N), where N is the number of non-zero elements. This is a big deal for practical applications.

 

[blue_raver22]

(n+1) - un(n)

As a scientist, my expertise may not be in the specific area of undergraduate math and PDEs, but I can offer some general advice and suggestions.

Firstly, it is important to have a good understanding of the underlying mathematical concepts and principles behind the problem you are trying to solve. In this case, it seems like you are dealing with a system of partial differential equations (PDEs) and are trying to code an implicit scheme for solving them. Implicit schemes are numerical methods used to approximate solutions to PDEs, and they can be quite complex to code.

My first recommendation would be to seek out resources that can help you better understand the theory and principles behind implicit schemes for PDEs. There are many textbooks, online courses, and tutorials available that can provide a solid foundation for your coding efforts.

In addition, it would be helpful to have a good understanding of the programming language you are using. Make sure you are familiar with the syntax and have a good grasp of the tools and libraries available to help with numerical computations.

As for the specific code you have provided, it looks like you are attempting to use a finite element method (FEM) for discretization. This is a common approach for solving PDEs numerically. However, it may be beneficial to consult with a mathematics or engineering professor or colleague who has experience with FEM to ensure your implementation is correct.

Overall, my advice would be to continue learning and seeking out resources to improve your understanding of the underlying mathematics and coding principles involved. With persistence and dedication, you will be able to successfully code an implicit scheme for your PDEs. Good luck!