# Numerical method to solve PDE system $\mathbf{A}\dot{\mathbf{U}} + \mathbf{B}\mathbf{U}^{'} + F(\mathbf{U}) = 0$

I would like to numerically solve a system of PDEs of the form

$$\mathbf{A}\dot{\mathbf{U}} + \mathbf{B}\mathbf{U}^{'} + F(\mathbf{U}) = 0$$

Where, $$\dot{\mathbf{U}}(s, t)$$ represents the time derivative, $$\mathbf{U}^{'}(s, t)$$ represents the space derivative, $$\mathbf{U}$$ is the vector of unknowns, and $$F(\mathbf{U})$$ is a nonlinear function of $$\mathbf{U}$$.

Both $$\mathbf{A}$$, and $$\mathbf{B}$$ are constant singular matrices.

So, the given system is a coupled system of PDEs and ODEs.

How to numerically solve such system of PDEs given initial and boundary conditions?. Which numerical method will be suitable?

• You probably will have to perform index reduction like in a system of DAE. Which it becomes if you disretize the space coordinate aka method-of-lines. Commented Feb 24, 2021 at 10:03

There are several related posts on this site where a possible and convenient method is presented. A possible approach known as operator splitting consists in writing two PDE systems \begin{aligned} &(S_1):\qquad {\bf A}\dot{\bf U} + {\bf B} {\bf U}' = 0 \\ &(S_2):\qquad {\bf A}\dot{\bf U} + F({\bf U}) = 0 \end{aligned} which are then integrated successively. The splitting error resulting from this process is often small enough compared to the numerical errors corresponding to the integration of $$S_1$$ and $$S_2$$, see this post. For the numerical resolution of $$S_1$$ a finite-difference method could be used, or a finite-volume scheme for instance. For the numerical resolution of $$S_2$$, a well-suited ODE solver will do. However, for most schemes the matrix $$\bf A$$ is required non-singular. One may need to introduce a change of variable or a perturbation of the initial system to rewrite it in a non-singular form. The most appropriate approach is very likely to depend on the form of the matrices and functions involved.