A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations

In this article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is based on a formulation of the discretized equations in vector form using only matrix-vector products and component-wise operations. By applying simple matrix-based differentiation rules, the matrix form of the analytical Jacobian can be calculated with little more difficulty than required to compute derivatives in single-variable calculus. After describing the direct matrix method, we present numerical experiments demonstrating the computational performance of the method, discuss its connection to the Newton–Kantorovich method and apply it to illustrative 1D and 2D example problems from electrochemical transport.