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Re: discretizing laplacian
Posted:
Feb 26, 2013 8:59 AM
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Sandeep Kumar <searchsandy1712@gmail.com> writes:
>Can anybody please tell me how to discretize Laplacian. I have -
>laplacian(Un^2.Un+1), which is laplacian of product of Un^2 and Un+1.
>where Un is U(j,n) and Un+1 is U(j,n+1).
>
> Thanks in advance.
>Regards
this makes no sense for me: with U already discretized, what should
laplacian of this mean?
If I remember right you one had laplacian(u^3) plus Crank-Nicholson,
that means you need
1/2*(laplacian(u(x,t_n)^3)+laplacian(u(x,t_{n+1})^3))
where of course laplacian is w.r.t x and -fortunately- x is onedimensional.
method 1:
(d/dx)^2(u(x,t_n)^3)(x=x_i) = 3u(i,n)^2(u(i+1,n)-2u(i,n)+u(i-1,n))/h^2
+(6/h^2)(u(i+1,n)-u(i-1,n))^2
method2:
(d/dx)^2(u(x,t_n)^3) =
(1/h)^2 ( ((u(i+1,n)^2+u(i,n)^2)/2)*(u(i+1,n)-u(i,n))
- ((u(i,n)^2+u(i-1,n)^2)/2)*(u(i,n)-u(i-1,n)) )
hth
peter
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