A Direct Matrix Method for Computing Analytical Jacobians of Discretized Nonlinear Integro-differential Equations

In this pedagogical 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 that required when computing 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. MATLAB code is provided to demonstrate the low code complexity required by the method.