Finite Differences-Semi discretization method on Heat Equation

In summary, the speaker is working on a personal project to solve the heat equation using their own Mathematica code. They are having trouble with a variable and are seeking help for a more intuitive animation. They are using semi discretization to discretize space and solving the differential equations for time without any numerical method. They are using Mathematica for its accuracy in solving the equations.
  • #1
Juan Carlos
22
0
Hi!, I'm working on a personal project: Solve the heat equation with the semi discretization method, using my own Mathematica's code, (W. Mathematica 9). The code:
View attachment PDE heat.nb

I'm having problems with the variable M (the number of steps). It works with M=1-5, but no further, I do not know what's going on. Help!
Also, I'm looking for a better animation more intuitive.
Thanks a lot
 
Physics news on Phys.org
  • #2
Since I do not have mathematica( using MATLAB for simulation purposes) I am just going to ask silly questions :|. As far as a quick google search is concerned Semi discretization means you only discretize space, right? And then you figure out some time evolution method. What are you using for the time evolution?
 
  • #3
Strum said:
Since I do not have mathematica( using MATLAB for simulation purposes) I am just going to ask silly questions :|. As far as a quick google search is concerned Semi discretization means you only discretize space, right? And then you figure out some time evolution method. What are you using for the time evolution?

Yes, that's the idea of semi discretization(also you can discretize the time or space)- But in this treatment I am working with the spatial variable (discrete), and solving the differential equations of time without any numerical method (certain boundary conditions and initial condition allow me to do it), since mathematica does it exactly.

Did I get you?
Thanks
 

1. How does the finite differences-semi discretization method work in solving the heat equation?

The finite differences-semi discretization method is a numerical method used to solve partial differential equations, such as the heat equation. It involves dividing the continuous domain into discrete points and approximating the derivatives with finite difference equations. By solving the resulting system of equations, the temperature distribution at each discrete point can be calculated, providing an approximation of the continuous solution to the heat equation.

2. What are the advantages of using the finite differences-semi discretization method?

One advantage of this method is that it is relatively easy to implement and can provide accurate solutions for a wide range of problems. It also allows for flexible discretization of the domain, meaning that the number of discrete points can be adjusted to balance computational efficiency and accuracy. Additionally, the method can handle complex boundary conditions and can be extended to solve other types of partial differential equations.

3. Are there any limitations to using the finite differences-semi discretization method?

The accuracy of the method depends on the size of the discretization, with smaller intervals leading to more accurate solutions but also increasing computational time. The method is also limited to domains with regular geometries and may not be suitable for solving problems with discontinuous or highly non-linear boundary conditions.

4. How does the finite differences-semi discretization method compare to other numerical methods for solving the heat equation?

Compared to other methods such as finite element or spectral methods, the finite differences-semi discretization method may be less accurate for complex problems. However, it is often faster and easier to implement, making it a popular choice for many applications. The choice of method will depend on the specific problem being solved and the desired balance between accuracy and computational efficiency.

5. Can the finite differences-semi discretization method handle time-dependent problems?

Yes, the method can be extended to handle time-dependent problems by using a time-stepping approach. The heat equation can be discretized in time as well as space, resulting in a system of equations that can be solved at each time step. This allows for the simulation of heat transfer over time, providing information on how the temperature distribution changes over time. However, this approach may require more computational resources compared to solving the steady-state heat equation.

Similar threads

B Incorporating boundary conditions in the Finite Element Method (FEM)
    • Differential Equations
Replies
4
Views
763
I 9-point Laplacian stencil derivation
    • Differential Equations
Replies
12
Views
5K
I Finite Element Method: Weak form to Algebraic Equations?
    • Differential Equations
Replies
1
Views
2K
Matlab report help please (Finite Difference Method)
    • Engineering and Comp Sci Homework Help
Replies
5
Views
1K
I Solving 2D Heat Equation w/ FEM & Galerkin Method
    • Differential Equations
Replies
7
Views
2K
A Crank-Nicholson solution to the cylindrical heat equation
    • Differential Equations
Replies
23
Views
4K
A PDE discretization for semi-infinite boundary?
    • Differential Equations
Replies
3
Views
1K
A How to find the partial derivatives of a composite function
    • Differential Equations
Replies
1
Views
2K
Nonlinear heat equation -- Handling the conductivity
    • Differential Equations
Replies
19
Views
2K
I Solving Poisson's Equation Using Finite Difference
    • Differential Equations
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top