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Water is one of the most biologically and economically important substances on Earth. A significant portion of Earth's water subsists in the subsurface. Our ability to monitor the flow and transport of water and other fluids through this unseen environment is crucial for a myriad of reasons.
One difficulty we encounter when attempting to model such a diverse environment is the nonlinearity of the partial differential equations describing the complex system. In
order to overcome this adversity, we explore Newton's method and its variants as feasible numerical solvers. The nonlinearity of the model problem is perturbed to create a linearized one. We then investigate
whether this linearized version is a reasonable replacement.
Finite difference methods are used to approximate the partial differential equations. We assess the appropriateness of
approximating the analytical Jacobian that arises in Newton's method, with an approximation method, to handle the event when certain derivative information is not available. |
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