Assume I have the following function:
$$0=a(\mathbf{x})h(\mathbf{x})+b(\mathbf{x})\nabla h(\mathbf{x})+c(\mathbf{x})$$
where $\mathbf{x}\in\mathbb{R}^{d}$ are points in some $d$-dimensional space, $a$, $b$, and $c$ are known functions $\mathbb{R}^{d}\rightarrow\mathbb{R}$. If it helps, we can assume that those functions are discretized on some form of grid. $\nabla$ is a sum of partial derivatives, i.e., $\nabla=\partial/\partial x_{1}+...+\partial/\partial x_{d}$.
I would like to solve for the function $h$ (or find its values on a grid). This would be trivial without the term $\nabla h(\mathbf{x})$, as I could simply identify local solutions at any point $\mathbf{x}$ I am interested in. However, its presence makes everything a lot more difficult. Do you know of any strategies to solve such problems?
I would also appreciate any recommendations for relevant literature or keywords to help me delve deeper into possible solution strategies! (I also apologize if my tags or title are imprecise - I would be grateful for any advice on how to specify my inquiry.)