Discretization scheme for cross derivative with time

General
1

Hi everyone,

I am modeling a PDE where the cross derivative with respect to time is involved. Specifically, the term of interest is of the form

\nabla\left(\frac{\partial Y}{\partial t}\right)

where Y is the variable to be solved. Currently, I approximate the term as

\nabla\left(\frac{\partial Y}{\partial t}\right)=\frac{\partial \nabla Y}{\partial t}\approx\frac{\nabla Y^{n+1}-\nabla Y^n}{\Delta t}

and it works, but the solution accuracy is not satisfactory. I am looking for a better discretization scheme for this type of cross derivative. Any help is greatly appreciated. Thanks.

2

Could you perform integration by parts, to get the derivative on the testfunction? You’d get some dYdt boundary conditions, or some interesting Robin-type weak conditions. Maybe you can somehow leverage those?

3

Thanks for the suggestion, but this term appears in my weak form as

\nabla v\cdot \nabla \left(\frac{\partial Y}{\partial t}\right)

and performing integration by parts will gives terms with \nabla^2v, more complicated than before.
For now I manage to increase the accuracy to match analytical results by using a Crank-Nicolson method.