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2D linear advection equation

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Old   February 11, 2011, 11:00
Default 2D linear advection equation
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Miguel Caro
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Well, trying to solve a 2D linear advection equation
u_t + au_x + bu_y = 0;
u_0(x,y,0) = sin( 2pi* x ) sin( 2pi y), (x,y) 0,1) x (0x1) , periodic boundary conditions

with exact solutions u(x,y,t) = sin (2pi (x-t) ) sin (2pi (y-t) )

i implemented this discretization :

u_i,j^{n+1} = u_i,j^n - dt/dx(Fi+1/2 - Fi-1/2) - dt/dy(Gi+1/2 - Gi-1/2);

CFL = max(dt/dx;dt/dy);

But the solution did not agree with the exact solution at a time t=2.0

Does anybody has a suggestion of what kind of algorithm to implement?

I would like to imlement a new-high resoluton scheme and i woud like to evaluate the convergence order. For this i would lik e to implement a technique including a flux limiter.

I realyy appreciate any help

Thanks
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Old   February 11, 2011, 12:31
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Try to increase the order of your temporal discretization by using a Runge-Kutta method (order 4 should do). This will allow you to use a reasonable time step and to obtain a more precise solution. Or, you may consider using an implicit temporal discretization.

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Old   February 11, 2011, 13:30
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Thanks Do, well i implemented yet a third order temporal RK, but with a more refined grid the numerical solution has an acceptable approximation to the reference solution. But I think i have to implement a fourth order temporal RK (usually it is called Strong Stability Prerserving Runge-Kutta of fourth order, for example. Gottlieb, Shu and others) to obtain better approximation.

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