Where to get started with Numerical Solutions to PDEs?

In summary, the conversation discussed the need to learn about numerical solutions for a nonlinear elliptic system of PDE's. The individual has some programming background and understanding of ODE numerical schemes but is unsure if they need to read books on Numerical Analysis or if there is a more direct way to learn. It was suggested to look into Finite Element techniques and books that focus on programming languages for this specific problem. Additionally, searching online for resources on nonlinear PDEs can also be helpful.
  • #1
lmedin02
56
0
I have established existence and uniqueness of solutions to a nonlinear elliptic system of PDE's and would like to see how the solutions look like numerically. I have some programming background on MatLab and C++. I also understand basic ODE numerical schemes. But I am not so sure, if I need to begin by first reading books on Numerical Analysis or is there a more direct way to learning about numerical solutions to my particular problem?

Thanks in advance for your discussion.
 
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  • #2
I'm not an expert on numerical methods for PDE's but since you haven't received other replies, I'll suggest that you look at material on "Finite Element" techiques if they apply to your PDE's. The Finite Element books I have glanced at (such as the Shaum's book) are very concrete. You can probably find books that do Finite Element analysis in particular programming languages. I notice the wording in one of the Shaum's books implies that older books on Finite Elements are obsolete because of modern developments. I don't understand any details about that, but it wouldn't hurt to get a modern book.
 
  • #3
Google 'nonlinear PDE'.
Let your fingers do the walking. There are several references to the numerical solution to nonlinear PDEs on the first page alone.
 

FAQ: Where to get started with Numerical Solutions to PDEs?

What are numerical solutions to PDEs?

Numerical solutions to PDEs (partial differential equations) are methods used to approximate solutions to these types of equations using numerical algorithms. PDEs are equations that involve multiple variables and their partial derivatives, and they are commonly used to model physical phenomena in various fields such as engineering, physics, and economics.

Why are numerical solutions necessary for solving PDEs?

Unlike ordinary differential equations, PDEs often do not have explicit solutions that can be solved algebraically. Therefore, numerical solutions are necessary to approximate the solutions to these complex equations. Additionally, numerical solutions allow for more accurate and efficient calculations compared to analytical methods.

What are some common numerical methods used for solving PDEs?

Some common numerical methods for solving PDEs include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the PDE into a system of algebraic equations, which can then be solved numerically using various algorithms.

How do I choose the appropriate numerical method for a specific PDE?

The choice of numerical method for a specific PDE depends on various factors such as the type of PDE, the boundary conditions, and the desired accuracy. It is important to carefully consider these factors and consult with experts in the field to determine the most suitable numerical method for a particular problem.

What are some challenges in using numerical methods for solving PDEs?

Some challenges in using numerical methods for solving PDEs include the need for a fine grid resolution to achieve high accuracy, the potential for numerical instability, and the computational cost of solving large systems of equations. Additionally, the choice of numerical method and the implementation of the algorithm require expertise and careful consideration to ensure accurate results.

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