Difficulties in the numerical solution of the partial differential equations governing seawater intrusion in aquifers arise from the coupling between the flow and transport equations and from the nonlinear aspects of this coupling. Several linearization approaches are discussed for the solution of the nonlinear system which results from a finite element discretization of the coupled equations. It is first shown that the most commonly used solution method can be viewed as a Picard linearization applied to the transport equation, with the coupling resolved by iteration over the two governing equations. The full Newton scheme for solving the coupled problem produces a Jacobian of size 2N x 2N, where N is the number of nodes in the discretization of both the flow and transport equations. To reduce the size and complexity of the full Newton scheme, a partial Newton method is proposed, which, like the Picard approach, produces matrix systems of size N x N. This scheme applies Newton linearization to the transport equation, and conventional iteration to resolve the coupling. Results from two- and three-dimensional test simulations show that the partial Newton scheme gives improved convergence and robustness compared to Picard linearization, especially for highly advective problems or large density ratios. Both approaches involve the solution of a symmetric (flow) and a nonsymmetric (transport) system of equations, and it is shown that the per iteration CPU cost for the partial Newton method is not significantly greater than that of the Picard scheme.

### Picard and Newton Linearization For the Coupled Model of Saltwater Intrusion In Aquifers

#### Abstract

Difficulties in the numerical solution of the partial differential equations governing seawater intrusion in aquifers arise from the coupling between the flow and transport equations and from the nonlinear aspects of this coupling. Several linearization approaches are discussed for the solution of the nonlinear system which results from a finite element discretization of the coupled equations. It is first shown that the most commonly used solution method can be viewed as a Picard linearization applied to the transport equation, with the coupling resolved by iteration over the two governing equations. The full Newton scheme for solving the coupled problem produces a Jacobian of size 2N x 2N, where N is the number of nodes in the discretization of both the flow and transport equations. To reduce the size and complexity of the full Newton scheme, a partial Newton method is proposed, which, like the Picard approach, produces matrix systems of size N x N. This scheme applies Newton linearization to the transport equation, and conventional iteration to resolve the coupling. Results from two- and three-dimensional test simulations show that the partial Newton scheme gives improved convergence and robustness compared to Picard linearization, especially for highly advective problems or large density ratios. Both approaches involve the solution of a symmetric (flow) and a nonsymmetric (transport) system of equations, and it is shown that the per iteration CPU cost for the partial Newton method is not significantly greater than that of the Picard scheme.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/2483972
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