A Direct Matrix Method for Computing Analytical Jacobians of Discretized Nonlinear Integro-differential Equations
<|reference_start|>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.<|reference_end|>
@article{chu2007a,
    title={A Direct Matrix Method for Computing Analytical Jacobians of Discretized
  Nonlinear Integro-differential Equations},
    author={Kevin T. Chu},
    journal={arXiv preprint arXiv:math/0702116},
    year={2007},
    doi={10.1016/j.jcp.2009.04.031},
    archivePrefix={arXiv},
    eprint={math/0702116},
    primaryClass={math.NA cs.CE}
}