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There is a difference between the requirements for a hyperbolic pde like $$ u_t + a u_x = 0 $$ and for a purely parabolic pde like $$ u_t = u_{xx} $$ Suppose the solutions are smooth and you approximate them by some finite difference method. Then in case of hyperbolic problem, the maximum error in the numerical solution depends on the time interval of ...


2

There seem to be some mistakes in your equation. You use y in some place and z in others. I suppose you are dealing with 2d flow. You need a maximum principle. At the PDE level, the maximum principle holds if the velocity is divergence-free. Assuming a divergence free velocity field, write an upwind scheme for convection terms and central scheme for ...


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