A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation

Author:

Berardi Marco1,Difonzo Fabio V.2

Affiliation:

1. Istituto di Ricerca sulle Acque, Consiglio Nazionale delle Ricerche, Via F. De Blasio 5, 70132 Bari, Italy

2. Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

Abstract

<p style='text-indent:20px;'>In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be <inline-formula><tex-math id="M1">\begin{document}$ l^{2} $\end{document}</tex-math></inline-formula>-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Medicine,Computational Mathematics,Computational Mechanics

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