How can I formulate spectral methods for numerical computation?

In summary, the conversation discussed the request for a reference showing examples of formulating spectral methods for computational problems. One suggestion was the book Computational Fluid Mechanics and Heat Transfer by Pletcher, Tannehill, and Anderson, which includes a section on spectral methods. The person asking for the reference also asked for the specific version and chapter, which was not readily available, and another resource, Numerical Recipes, was mentioned.
  • #1
RobosaurusRex
29
1
Hi,

I am new to spectral methods, does anyone have a good reference which shows some worked examples of formulating the method to be computed?
E.g. go from the general governing equations through transformation to the form which will be passed to a computer.

Cheers
 
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  • #2
One of my go-to references for computational methods is Computational Fluid Mechanics and Heat Transfer by Richard H. Pletcher, John C. Tannehill, Dale Anderson. The book shows derivations and applications to different types of mass transfer problems and has a section on spectral methods. Maybe you can find a copy at your library.
 
  • #3
NFuller said:
One of my go-to references for computational methods is Computational Fluid Mechanics and Heat Transfer by Richard H. Pletcher, John C. Tannehill, Dale Anderson. The book shows derivations and applications to different types of mass transfer problems and has a section on spectral methods. Maybe you can find a copy at your library.

Can you tell me which version of the book and which chapter you are referring to?
I have third edition and I cannot find spectral methods in the glossary.

Thanks for the quick reply.
 
  • #4
I also have the fourth edition but I don't have the book on hand right now. From what I remember is was talked about in the Application of numerical methods to selected model equations. If it's not in there, another good resource on this is chapter 13 of Numerical Recipes.
 

FAQ: How can I formulate spectral methods for numerical computation?

What is spectral method formulation?

Spectral method formulation is a mathematical framework used in scientific research to analyze and solve problems that involve continuous functions. It involves representing a function as a linear combination of basis functions, such as polynomials or trigonometric functions, and using numerical techniques to approximate the solution.

How is spectral method formulation different from other numerical methods?

Spectral methods are different from other numerical methods, such as finite difference or finite element methods, in that they use a global representation of the function being analyzed. This allows for a more accurate and efficient approximation of the solution, particularly for smooth functions or those with periodic behavior.

What are the benefits of using spectral method formulation?

There are several benefits to using spectral methods, including high accuracy and efficiency for smooth functions, fast convergence rates, and the ability to handle problems with periodic boundary conditions. They also have the advantage of being applicable to a wide range of problems, including differential equations, optimization, and image processing.

What are the limitations of spectral method formulation?

While spectral methods have many advantages, they also have some limitations. They are not suitable for problems with discontinuities or sharp gradients, and they may require a large number of basis functions to accurately represent complex functions. They also may be more computationally expensive compared to other numerical methods.

How is spectral method formulation used in scientific research?

Spectral methods have a wide range of applications in scientific research, particularly in fields such as fluid dynamics, astronomy, and numerical analysis. They are used to solve differential equations, simulate physical processes, and analyze data. They are also commonly used in conjunction with other numerical methods to improve accuracy and efficiency.

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