Is the Half Step Scheme Truly TVD? Help Needed to Prove It

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In summary, TVD stands for total variation diminishing and is a property of numerical schemes used to solve hyperbolic partial differential equations. A half step scheme is a type of numerical method that can be used to solve these equations, and it involves splitting the problem into two subproblems. To prove that a half step scheme is TVD, one must show that it satisfies the TVD condition and this can be done using the discrete maximum principle and Lax-Wendroff theorem. The advantages of using a half step scheme include its simplicity, stability, and adaptability to higher dimensions. However, it also has limitations such as potential inaccuracies and computational expenses. The choice of splitting the problem can also affect the overall accuracy and stability of the
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martizzle
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Hi,

I'm trying to prove a half step scheme is TVD and I'm stuck. I wasn't sure how to type it all here so I typed it up in latex, see attached pdf for the problem. Any tips would be much appreciated.

Thanks.
 

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Thanks for the direction. I posted it there.
 
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In future posts, please expand acronyms such as TVD and CFL, which is used in the document you attached. I presume that the latter isn't short for Canadian Football League or Compact Fluorescent Light.
 
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1. What is TVD?

TVD stands for total variation diminishing and is a property of numerical schemes used to solve hyperbolic partial differential equations. It ensures that the numerical solution does not produce spurious oscillations or non-physical behavior.

2. How do you prove that a half step scheme is TVD?

To prove that a half step scheme is TVD, one must show that it satisfies the TVD condition, which states that the total variation of the numerical solution at a given time step is less than or equal to the total variation of the initial data. This can be done by applying the discrete maximum principle and using a suitable numerical analysis technique such as the Lax-Wendroff theorem.

3. What is a half step scheme?

A half step scheme is a numerical method used to solve hyperbolic partial differential equations. It involves splitting the original problem into two subproblems, solving them separately, and then combining the solutions to obtain the final numerical solution. The half step scheme is often used for time-dependent problems and can be applied to a wide range of hyperbolic equations.

4. What are the advantages of using a half step scheme?

One of the main advantages of using a half step scheme is that it is relatively simple to implement and can be applied to a variety of problems. It also has good stability properties and can handle discontinuities or shocks in the solution. Additionally, it can be easily adapted to higher dimensions and can be combined with other numerical methods to improve accuracy.

5. Are there any limitations to using a half step scheme?

Like any numerical method, a half step scheme also has its limitations. It may not be suitable for all types of problems and may not always produce accurate results. It can also be computationally expensive for certain problems. Additionally, the choice of splitting the original problem into two subproblems can also affect the overall accuracy and stability of the scheme.

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