Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution

In summary, the conversation discusses the use of Comsol 5.3 to numerically solve a partial differential equation with specific boundary conditions and initial conditions. The equation includes a parameter, St, and the person is interested in tracking the evolution of a specific term involving St over time. However, there is concern about the correctness of the problem due to a term that cannot be a function of the unknown variable. Further study is needed to determine the correctness of the problem.
  • #1
umby
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TL;DR Summary
Numerical solution of a partial differential equation containing the derivative of the unknown at a point
What is the best way to solve numerically the following equation using Comsol 5.3.

##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}##
##T(0,t)=1##
##T(\infty ,t)=0##
##T(x,0)=\exp \left(-\frac{x^2}{\pi }\right)-x \text{erfc}\left(\frac{x}{\sqrt{\pi }}\right)##

where ##\text{St}## is a parameter which can varies from 0.01 to 100.
I am particularly interested in following the evolution of ##\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]## with time.
Thanks in advance.
 
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  • #2
Be care. This is not a PDE at all. Even the correctness of such a problem needs for a separate study.
 
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  • #3
wrobel said:
Be care. This is not a PDE at all. Even the correctness of such a problem needs for a separate study.
You mean because of the coefficient of the derivative of ##T## with respect to ##x##? Maybe differential relation is better? Can you help me please in determining the correctness of this problem?
 
Last edited:
  • #4
I mean the term
$$\frac{\partial T}{\partial x}\Big|_{x=0}$$
 
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  • #5
Exsactly what I was mentioning, thank you for point it out. Coefficients cannot be function of the unknown, only of the indipendent variables, in my case ##x## and ##t##.
Can you please tell me more about the matter of its correctness?
 
  • #6
You can probably write a finite difference code easier than using comsol. Then you can insert the nonlinear part of the PDE with ease.
 
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1. How do I set up and solve a PDE in Comsol 5.3?

To set up and solve a PDE in Comsol 5.3, you first need to define the geometry, physics, and boundary conditions of your problem. Then, you can create a new study, select the PDE interface, and specify the PDE coefficients and terms. Finally, you can solve the PDE using the built-in solvers in Comsol 5.3.

2. How can I numerically solve a PDE in Comsol 5.3?

To numerically solve a PDE in Comsol 5.3, you can use the finite element method (FEM) provided by the software. Comsol 5.3 automatically discretizes the PDE into a set of algebraic equations, which are then solved using iterative methods to obtain the numerical solution.

3. How do I visualize the time evolution of a PDE solution in Comsol 5.3?

To visualize the time evolution of a PDE solution in Comsol 5.3, you can create animations or plots of the solution over time. By specifying a time-dependent study and solving the PDE for different time steps, you can observe how the solution changes over time using the visualization tools in Comsol 5.3.

4. Can I customize the numerical solution settings in Comsol 5.3?

Yes, you can customize the numerical solution settings in Comsol 5.3 to improve the accuracy or efficiency of the solution. You can adjust parameters such as mesh refinement, solver type, convergence criteria, and time-stepping methods to tailor the numerical solution to your specific needs.

5. How do I validate the numerical solution of a PDE in Comsol 5.3?

To validate the numerical solution of a PDE in Comsol 5.3, you can compare it with analytical solutions, experimental data, or results from other numerical methods. By performing convergence studies, sensitivity analyses, or verification tests, you can ensure that the numerical solution obtained in Comsol 5.3 is accurate and reliable.

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