scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows

2011 ◽  
Vol 9 (1) ◽  
pp. 205-230 ◽  
Author(s):  
Angela Ferrari ◽  
Claus-Dieter Munz ◽  
Bernhard Weigand
Keyword(s):  
Finite Volume ◽  
High Order ◽  
Indicator Function ◽  
Finite Volume Scheme ◽  
Riemann Solver ◽  
Time Step ◽  
Material Interface ◽  
Volume Scheme ◽  
Time Stepping ◽  
Local Time Stepping

AbstractIn this paper, a new sharp-interface approach to simulate compressible multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative discontinuous Galerkin finite element scheme to evolve an indicator function that tracks the material interface. At the interface our method applies ghost cells to compute the numerical flux, as the ghost fluid method. However, unlike the original ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived from an approximate-state Riemann solver, similar to the approach proposed in [25], but based on a much simpler formulation. Our formulation leads only to one single scalar nonlinear algebraic equation that has to be solved at the interface, instead of the system used in [25]. Away from the interface, we use the new general Osher-type flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann solver, applicable to general hyperbolic conservation laws. The time integration is performed using a fully-discrete one-step scheme, based on the approaches recently proposed in [5,7]. This allows us to evolve the system also with time-accurate local time stepping. Due to the sub-cell resolution and the subsequent more restrictive time-step constraint of the DG scheme, a local evolution for the indicator function is applied, which is matched with the finite volume scheme for the solution of the Euler equations that runs with a larger time step. The use of a locally optimal time step avoids the introduction of excessive numerical diffusion in the finite volume scheme. Two different fluids have been used, namely an ideal gas and a weakly compressible fluid modeled by the Tait equation. Several tests have been computed to assess the accuracy and the performance of the new high order scheme. A verification of our algorithm has been carefully carried out using exact solutions as well as a comparison with other numerical reference solutions. The material interface is resolved sharply and accurately without spurious oscillations in the pressure field.

scholarly journals Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations

2020 ◽  
Vol 30 (13) ◽  
pp. 2487-2522
Author(s):  
Rafael Bailo ◽  
José A. Carrillo ◽  
Hideki Murakawa ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Flow Structure ◽  
Gradient Flow ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Fully Discrete ◽  
Local Aggregation

We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502 ]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

Numerical Analysis of a Stable Finite Volume Scheme for a Generalized Thermistor Model

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mustapha Ghilani ◽  
El Houssaine Quenjel ◽  
Mohamed Rhoudaf
Keyword(s):  
Weak Solution ◽  
Finite Volume ◽  
Electric Potential ◽  
Discrete Maximum Principle ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Stiffness Coefficients ◽  
Fully Implicit ◽  
The Stability ◽  
Stability Issues

AbstractA generalized thermistor model is discretized thanks to a fully implicit vertex-centered finite volume scheme on simplicial meshes. An assumption on the stiffness coefficients is mandatory to prove a discrete maximum principle on the electric potential. This property is fundamental to handle the stability issues related to the Joule heating term. Then the convergence to a weak solution is established. Finally, numerical results are presented to show the efficiency of the methodology and to illustrate the behavior of the temperature together with the electric potential within the medium.

scholarly journals Convergence of a finite volume scheme for a system of interacting species with cross-diffusion

2020 ◽  
Vol 145 (3) ◽  
pp. 473-511 ◽  
Author(s):  
José A. Carrillo ◽  
Francis Filbet ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Coupled System ◽  
Convergence Result ◽  
The Self ◽  
Finite Volume Scheme ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Interacting Species

Abstract In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

Computation of Sub-Micron Thermal Transport Using an Unstructured Finite Volume Method

2002 ◽  
Vol 124 (6) ◽  
pp. 1176-1181 ◽  
Author(s):  
J. Y. Murthy ◽  
S. R. Mathur
Keyword(s):  
Finite Volume ◽  
Isotropic Scattering ◽  
Finite Volume Scheme ◽  
Boltzmann Transport ◽  
Linear Set ◽  
Volume Scheme ◽  
Relaxation Time Approximation ◽  
Fully Implicit ◽  
Volume Method ◽  
Unstructured Finite Volume Method

An unstructured finite volume scheme is applied to the solution of sub-micron heat conduction problems. The phonon Boltzmann transport equation (BTE) in the relaxation time approximation is considered. The similarity between the radiative transfer equation (RTE) and the BTE is exploited in developing a finite volume scheme for the BTE. The spatial domain is divided into arbitrary unstructured polyhedra, the angular domain into control angles, and the frequency domain into frequency bands, and conservation equations for phonon energy are written. The unsteady wave propagation term, not usually present in thermal radiation problems, is differentiated using a fully implicit scheme. A sequential multigrid scheme is applied to solve the nominally linear set. Isotropic scattering due to a variety of mechanisms such as impurity and Umklapp scattering is considered. The numerical scheme is applied to a variety of sub-micron conduction problems, both unsteady and steady. Favorable comparison is found with the published literature and with exact solutions.

Computation of Sub-Micron Thermal Transport Using an Unstructured Finite Volume Method

2001 ◽  
Author(s):  
J. Y. Murthy ◽  
S. R. Mathur
Keyword(s):  
Finite Volume ◽  
Isotropic Scattering ◽  
Finite Volume Scheme ◽  
Boltzmann Transport ◽  
Linear Set ◽  
Volume Scheme ◽  
Relaxation Time Approximation ◽  
Fully Implicit ◽  
Volume Method ◽  
Unstructured Finite Volume Method

Abstract An unstructured finite volume scheme is applied to the solution of sub-micron heat conduction problems. The phonon Boltzmann transport equation (BTE) in the relaxation time approximation is considered. The similarity between the radiative transfer equation (RTE) and the BTE is exploited in developing a finite volume scheme for the BTE. The spatial domain is divided into arbitrary unstructured polyhedra, the angular domain into control angles and the frequency domain into frequency bands and conservation equations for phonon energy are written. The unsteady wave propagation term; not usually present in thermal radiation problems, is differenced using a fully implicit scheme. A sequential multigrid scheme is applied to solve the nominally linear set. Isotropic scattering due to a variety of mechanisms such as impurity and Umklapp scattering is considered. The numerical scheme is applied to a variety of sub-micron conduction problems, both unsteady and steady. Favorable comparison is found with the published literature and with exact solutions.

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