# (Algorithmic) Differentiation capable Finite Volume Software: Generation Jacobian

I am looking for Finite Volume Software that employs a method-of-lines like approach by constructing from the hyperbolic PDE of form $$\partial_t \boldsymbol u(t,\boldsymbol x) + \nabla \cdot \boldsymbol f\big(\boldsymbol u(t,\boldsymbol x)\big) = \boldsymbol 0 \tag{1}$$ the semidiscretized ODE $$\frac{d }{d t} \boldsymbol U(t) = \boldsymbol F\big(\boldsymbol U(t)\big) \tag{2} \label{2}$$ and allows the computation of the Jacobian of the RHS of \eqref{2}:

$$J(\boldsymbol u) = \nabla_{\boldsymbol u} \boldsymbol F (\boldsymbol u)$$

Preferably, this is done with algorithmic/automatic differentiation, although finite differences should also be alright for the beginning. Also, computation for the initial value $$\boldsymbol u_0$$ would also suffice here.

I checked standard software like Clawpack and FiPy but they do not seem to offer such capabilities.

• If DG is an acceptable substitute then I can recommend things but I don't know anything for finite volume :/ Dec 20, 2022 at 15:28
• @Daniel Shapero Does your recommendation supports polynomial degrees $P=0$? Could be accepted as an answer. Dec 20, 2022 at 17:36
• I am particularly NOT asking for DG - although zero'th order polynomials would be a starting point Dec 20, 2022 at 23:43
• github.com/SciML/MethodOfLines.jl has PDE support in progress, but you also can always hand convert the system. It's not yet ideal for PDE's but it will give you AD for free. Dec 23, 2022 at 2:57
• MethodOfLines.jl does WENO schemes. Dec 25, 2022 at 19:14