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I want to simulate on my computer the solution of the heat equation in 3 space dimensions with Cauchy initial data, that is $$\partial_t u=Tr[A(x)\cdot \Delta u], u(0,x)=u_0(x) $$

where $u_0\in C(\mathbb{R}^3,\mathbb{R})$.

Even if $A$ is constant I'm not sure what's numerically the best way to do this but I'm sure this must be in standard toolboxes of Matlab, Scilab etc. however I have quite some trouble with the documentation.

Could someone give me pointers in the right direction? More general, what is a good book or lecture notes that deal with numerics for parabolic PDEs (with multidimensions in space)?

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The way the equation is written looks unusual. Also don't you need to specify the conditions at the boundary of the spatial domain? –  Dmitri Chubarov Jan 23 '13 at 18:16
1  
That's not the heat equation. Maybe you inserted an extra time derivative by mistake. –  David Ketcheson Jan 24 '13 at 11:22
    
What is $\Tr$ in this case? –  Martin Jan 26 '13 at 15:03
    
Hi john. As your question currently stands, there is some critical information missing which prevents us from answering it. As @DmitriChubarov mentions, boundary conditions are required. Also, it is unclear what you mean by $Tr$. Please update your question with the necessary information. –  Paul Sep 5 '13 at 14:04
    
By Tr I just mean the Trace of the diffusion matrix A multiplied with the Hessian, i.e. it's equivalent to summing the second derivatives and multiplying each with a space dependent diffusion constant. Considering boundary conditions: above equation is is well-posed and the point is exactly that I have an equation on an unbounded domain and find numerics which work there. –  john Oct 2 '13 at 9:37
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