MATLAB Solving 2nd Order PDE System with Crank-Nicholson

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SUMMARY

This discussion focuses on solving a system of second-order partial differential equations (PDEs) using the Crank-Nicholson (CN) method. The system includes nonlinear terms, particularly in the first equation involving the variable E, which complicates the application of the CN scheme. The user suggests employing the Newton-Raphson method to address the nonlinearity and seeks confirmation on the appropriateness of this approach. The conversation emphasizes the need for clarity in notation to avoid confusion when dealing with vector quantities.

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  • Understanding of second-order partial differential equations (PDEs)
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  • Knowledge of Newton-Raphson method for solving nonlinear equations
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  • Research the implementation of the Crank-Nicholson method for nonlinear PDEs
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Mathematicians, engineers, and researchers working on numerical methods for PDEs, particularly those interested in nonlinear systems and numerical analysis using MATLAB.

hunt_mat
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I have the following system of PDEs:
<br /> \hat{\rho}\hat{c}_{th}\frac{\partial\hat{T}}{\partial\hat{x}}-\alpha_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{k}(\hat{x})\frac{\partial\hat{T}}{\partial\hat{x}}\right)=\alpha_{1}\hat{\sigma}(\hat{x})\hat{E}<br />
<br /> \frac{\partial}{\partial\hat{x}}(\hat{\varepsilon}(\hat{x})\hat{E})=-\beta\hat{c}<br />
<br /> \frac{\partial\hat{c}}{\partial\hat{t}}-\gamma_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{D}(\hat{x})\frac{\partial\hat{c}}{\partial\hat{x}}\right)= \gamma_{2}\left(\frac{\partial\hat{E}}{\partial\hat{x}}+\frac{\partial\hat{c}}{\partial\hat{x}}-\frac{\partial\hat{T}}{\partial\hat{x}}\right)<br />

I would like to solve this system using the Crank-Nicholson method. Now for a linear equation, the CN scheme is well defined, MATLAB has some very nice algorithms for this.

However the first equation has a nonlinear term in E, and I have no equation which time steps E. I suppose that I could use a Newton-Raphson scheme to get the solution. Would that be the correct way forward?
 
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What is your question?
 
What would be the best way forward? As stated in my post.
 
hunt_mat said:
What would be the best way forward? As stated in my post.
Sorry about that.
For some reason no text after your first equation is visible in Safari on iOS.
 
a perturbation expansion for E (kill all the \hat{}, it makes the equations hard to read and is confusing, unless they are all vector quantities,l then you have a mess and an intractable system).
 

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