A conservative flux for the continuous Galerkin method based on discontinuous enrichment

Larson, Mats G.; Niklasson, A. Jonas

doi:10.1007/s10092-004-0084-7

Cited by 28 publications

(25 citation statements)

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“…To rectify this we have implemented a post-processing technique given in [64] and applied to subsurface unsaturated flow in [63]. For the incompressible Navier-Stokes equations the approach yields locally divergence-free, optimally accurate velocities on element boundaries, and we then project the resulting velocities to element interiors with appropriate local representations such as the lowest order Raviart-Thomas space [65] or the Brezzi-Douglas-Marini space of order one [66].…”

Section: Navier-stokes Equationmentioning

confidence: 99%

“…This global continuity property has sometimes incorrectly been assumed to imply the lack of a local conservation property. In fact mass is still redistributed through local numerical fluxes and post-processing can be used to define phase fluxes the are conservative on cells [64]. where / h is the approximate solution, / is the analytical solution, and p h (C) is the piecewise linear surface constructed from extracting the zero contour of the piecewise linear nodal projection of / onto the computational mesh with ParaView [74].…”

Section: Zalesak's Diskmentioning

confidence: 99%

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A conservative level set method suitable for variable-order approximations and unstructured meshes

Kees

Akkerman

Farthing

et al. 2011

Journal of Computational Physics

124

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Section: Navier-stokes Equationmentioning

confidence: 99%

Section: Zalesak's Diskmentioning

confidence: 99%

A conservative level set method suitable for variable-order approximations and unstructured meshes

Kees

Akkerman

Farthing

et al. 2011

Journal of Computational Physics

124

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“…Rather than computing a full DG solution, however, it is also possible to postprocess a nonconservative approximation so as to satisfy the compatibility conditions; see, for example, [22,33]. Consider the reference elementˆ defined in (ξ, η) space, pictured in Fig.…”

Section: The Cg Optionmentioning

confidence: 99%

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

Dawson

2008

Comput Geosci

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We consider conjunctive surface-subsurface flow modeling, where surface water flow is described by the shallow water equations and ground water flow by Richards' equation for the vadose zone. Coupling between the models is based on the continuity of flux and water pressure. Numerical approximation of the coupled model using the framework of discontinuous Galerkin (DG) methods is formulated. In the subsurface, the local discontinuous Galerkin (LDG) method is used to approximate ground water velocity and hydraulic head; a DG method is also used to approximate surface water velocity and elevation. This approach allows for a weak coupling of the models and the use of different approximating spaces and/or meshes within each regime. A simplified LDG method based on continuous approximations to water head is also described. Numerical results that investigate physical and numerical aspects of surface-subsurface flow modeling are presented.

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“…As a result, cell-centered finite differences and low-order finite volumes/ integrated finite differences, and piecewise linear, continuous Galerkin finite element schemes remain the dominant schemes used in practice (Selker and John, 2004;Kollet and Maxwell, 2006;Yeh et al, 2011;Diersch, 2013). We note that post-processing is increasingly being used in conjunction with continuous Galerkin methods to provide velocity fields that are locally conservative over elemental control volumes (Larson and Niklasson, 2004;Sun and Wheeler, 2006;Kees et al, 2008;Scudeler et al, 2016).…”

Section: Spatial Discretizationmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

249

138

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The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

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A conservative flux for the continuous Galerkin method based on discontinuous enrichment

Cited by 28 publications

References 0 publications

A conservative level set method suitable for variable-order approximations and unstructured meshes

A conservative level set method suitable for variable-order approximations and unstructured meshes

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

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