How to distribute gradient?

Question

I am using the finite difference method to solve a pde in time and 1D space. I am trying to distribute the gradient and want to make sure I am doing this correct.

For $d,a \in R$ and $u(x,t)$ and $c(x,t)$ I have

$\nabla \cdot ((d+a|u|)\nabla c) = \nabla \cdot (d\nabla c)+ \nabla \cdot (a|u|\nabla c) = \nabla^2 c + a\nabla|u|\nabla^2c$

Is this correct?

Answer 1

• Better use the Laplacian operator in $∇⋅∇c=div(grad(c))=\Delta c$ to make a difference from the Hessean.
• You can not get a product with 3 derivatives when the original construct has only nested derivatives two levels deep.
• Use the product rule $∇⋅(|u|∇c)=(∇|u|)⋅(∇c)+|u|(∇⋅∇c)$.