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

<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, resort Kirchhoff transformation of in mixed form, so exploit specific Gardner model features, obtaining linear second partial differential equation. Then, leveraging the balance condition, integrate both sides over generic grid cell discretize integrals using trapezoidal rule. This approach provides non-homogeneous initial value problem with respect transform variable, whose solution yields sought scheme. Such is proven be <inline-formula><tex-math id="M1">\begin{document}$ l^{2} $\end{document}</tex-math></inline-formula>-stable convergent exact under suitably conditions on step-sizes, retaining convergence from underlying quadrature formula.</p>