Solving Partial Differential Equation Systems Numerically

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In summary, the conversation revolves around the integration of a system of partial differential equations in "r" and "z" coordinates, specifically the continuity, momentum, and energy equations. The use of the Crank-Nicholson method in Matlab or Fortran is mentioned, and the speaker is seeking assistance in finding a suitable method and software for solving the problem. Other tools such as Mathematica and CFD are also brought up as alternatives but the speaker is most familiar with Matlab. The problem being discussed is related to deflagration initiation by hot jets and the speaker is looking for a website with pre-existing programs for CFD.
  • #1
Clausius2
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I have to integrate numerically the next system of partial differential equations, in "r" and "z" coordinates (steady state, cylindrical coordinates):
-Continuity equation
-Momentum equation (with viscous and convection terms)
-Energy equation (with heat transfer, convection, and internal generation terms).
I heard something about Crank-Nicholson method. I have to implement this problem in Matlab or Fortran.
Anybody know about how can I solve this heavy task?
Anybody know about some web where I could obtain a method for solving it, and implemented in the software specified?.

Thanks for answering.
 
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  • #2
Does it need to be in Matlab or Fortran, or are those the ones you know how to use?

It may be easier in Mathematica (my new favorite tool).
 
  • #3
To be honest, Matlab is my useful program. I've been working with it very usually. I don't know anything about Mathematica.
This is the heaviest problem I've never simulated numerically.It is a part of a study of "deflagration initiation by hot jets".
 
  • #4
I'm looking for a website holding worked programs of CFD. Do you know anyone?. I'm too lazy to do the program on my way!
 

FAQ: Solving Partial Differential Equation Systems Numerically

What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe physical phenomena that vary continuously over space and time.

How are partial differential equations solved numerically?

PDEs can be solved numerically by discretizing the problem domain into a grid and approximating the derivatives using finite difference methods. This results in a system of algebraic equations, which can be solved using numerical techniques such as iterative methods or direct solvers.

What are some common techniques for solving PDE systems numerically?

Some common techniques for solving PDE systems numerically include finite difference methods, finite element methods, and spectral methods. These methods vary in their accuracy and computational complexity, and the choice of method depends on the specific problem being solved.

What are some challenges in solving PDE systems numerically?

Solving PDE systems numerically can be challenging due to the complexity of the equations, the need for high computational power, and the potential for numerical instability. Additionally, the choice of discretization scheme and numerical parameters can greatly affect the accuracy and efficiency of the solution.

What are some applications of solving PDE systems numerically?

PDE systems are used in many fields, including physics, engineering, and finance. Some specific applications include modeling heat transfer, fluid dynamics, and option pricing in financial markets. Solving these systems numerically allows for more accurate and efficient predictions of real-world phenomena.

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