Suggestions for scheme to use to solve PDE numerically

In summary, the conversation discusses the process of modeling laser ablation on a material using MATLAB. The equation governing the process is a non-linear parabolic PDE and the individual is seeking guidance on how to proceed with solving it numerically. It is suggested to use the Crank-Nicolson method and to non-dimensionalize time and space to better understand the problem. The stability of the solution may also be affected by the chosen time and space increments. It is also mentioned that pdepe automatically chooses an appropriate time step for stability.
  • #1
Chim
5
0
Hello everyone,
I am trying to model the process of laser ablation on a material using MATLAB. The governing equation is of the form:
∂T(x,t)/∂t = ∂/∂x(A*∂T/∂x) + B*exp(-C*t2)*exp(-D*x)
with one Initial condition and two boundary conditions.
Using the built-in 'pdepe' function in Matlab gave inaccurate results, so I have been reading material on solving it numerically. This being a non-linear parabolic pde (correct me if wrong), I am facing difficulty on how to proceed. All the materials that I have referred to so far only address problems where the last term (source term) of the equation is a function of T(x,t). Please guide me.
 
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  • #2
Is A a function of x or t or T?

If its not a function of any of the above this is just a non-homogenous heat equation and you can solve it analytically.

If it's a function of x or t or both, it's still a linear equation.

If it's a function of T then indeed this is non-linear.
 
  • #3
Double Post. Sorry.
 
  • #4
Usually for these problems the Crank-Nicolson method suffices. Note that your source term is quite sensitive to t and x with an exponential decay so a very fine mesh is probably needed when x<<1 and t<<1 to capture the source term accurately.

Also, in numerical methods, it doesn't matter (much) if the source term depends on T or not. In this case, the source term is known on every point of the grid if B,C,D are not functions of T.

You should also be able to get the analytical solution as kai_sikorski mentioned. It is always good to have an analytical solution.
 
  • #5
Sorry. Forgot to mention about the coefficients earlier.
A is a function of T, whereas B,C and D are positive constants.

kai_sikorski said:
Is A a function of x or t or T?

If its not a function of any of the above this is just a non-homogenous heat equation and you can solve it analytically.

If it's a function of x or t or both, it's still a linear equation.

If it's a function of T then indeed this is non-linear.
 
  • #6
As you have noticed that i will be needing an extremely fine mesh, I was using a time step of 1ns and a 10nm step for distance x. Is it possible that using these increments might affect the stability of the solution?
I would also like to mention that I need to study the behaviour in the first 100ns and the ablation was observed till a few μm.

bigfooted said:
Usually for these problems the Crank-Nicolson method suffices. Note that your source term is quite sensitive to t and x with an exponential decay so a very fine mesh is probably needed when x<<1 and t<<1 to capture the source term accurately.

Also, in numerical methods, it doesn't matter (much) if the source term depends on T or not. In this case, the source term is known on every point of the grid if B,C,D are not functions of T.

You should also be able to get the analytical solution as kai_sikorski mentioned. It is always good to have an analytical solution.
 
  • #7
It might be a good idea to non-dimensionalize time and space. Quantities with units make it hard to tell what's large and what's small. Is 1ns small? Well it depends on how fast the forcing in the problem is varying. If C~(1 ns)^2, or smaller then your time step might be huge in comparison. So I would non-nimensionalize time by (C)^{-1/2}, and space by (D)^{-1}, that way the forcing is changing over order 1 quantities in space and time and the meaning of what a small time step is will be more clear.

Stability is usually related to the time step being too large compared to the space step. I just read the documentation for pdepe though and it seems that it actually chooses an appropriate time step on it's own, the time points you specify are just the time points it will give you a solution at, but they have nothing to do with what time step it actually used. So presumably it chooses something that should be stable.
 

What is a Partial Differential Equation (PDE)?

A PDE is a mathematical equation that involves multiple variables and their partial derivatives. It is used to describe physical phenomena such as heat transfer, fluid dynamics, and electrostatics.

Why do we need to solve PDEs numerically?

Many PDEs are too complex to be solved analytically, meaning that there is no exact solution. Therefore, numerical methods are used to approximate the solution and provide useful insights into the behavior of the system.

What is a numerical scheme for solving PDEs?

A numerical scheme is a set of algorithms and procedures used to approximate the solution of a PDE. It involves discretizing the PDE into smaller pieces and solving each piece numerically using techniques such as finite difference, finite element, or spectral methods.

What factors should be considered when choosing a numerical scheme for solving PDEs?

The choice of a numerical scheme depends on various factors such as the type of PDE, boundary conditions, stability, accuracy, computational efficiency, and ease of implementation.

What are some common numerical schemes used for solving PDEs?

Some common numerical schemes for solving PDEs include the finite difference method, finite element method, and spectral (Fourier, Chebyshev) methods. Each method has its own advantages and limitations, and the choice depends on the specific problem at hand.

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