About solving Transport PDE

[macrovue]

Here's my question, friends

I have to define initial and boundary condition for a transport PDE: u_t+x(1-x)u_x=0
with x and t is between [0,1], to solve this equation, what kind of numerical method
and boundary condition do you recommend and why?

What kind of numerical error do you expect?

Detailed explanation will be appreciated in advance.

 

[TaiwanCountry]

Hi , How do you do ?

you can try this answer.

u=f(lnx -ln(1-x)-t) ln() for natural log.

The method is here
http://en.wikipedia.org/wiki/Method_of_characteristics"


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ROC(Taiwan) is not PRC.

http://en.wikipedia.org/wiki/Taiwan"
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[Mene]

I would recommend using a finite difference method to solve this transport PDE. This method involves dividing the domain into a grid of discrete points and approximating the derivatives using finite differences. This method is commonly used for solving PDEs because it is relatively simple and accurate.

For the boundary conditions, I would recommend using a combination of Dirichlet and Neumann boundary conditions. Dirichlet boundary conditions specify the value of the solution at the boundary points, while Neumann boundary conditions specify the derivative of the solution at the boundary points. In this case, we could use Dirichlet boundary conditions at the initial time (t=0) to specify the initial condition, and Neumann boundary conditions at the boundaries (x=0 and x=1) to ensure the solution does not leave the domain.

In terms of numerical error, we can expect truncation error and round-off error. Truncation error is the error introduced by approximating the derivatives using finite differences, while round-off error is caused by the limitations of computer arithmetic. The amount of error will depend on the size of the grid and the accuracy of the computer used for the calculations. However, we can minimize these errors by choosing an appropriate grid size and using higher precision arithmetic if available.

I hope this explanation helps in solving your transport PDE and understanding the numerical methods and boundary conditions involved. If you have any further questions, please do not hesitate to ask.