Mass Conservative Scheme Research Articles

Stable and locally mass- and momentum-conservative control-volume finite-element schemes for the Stokes problem

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.

Robustness and efficiency of iteration schemes for variably saturated flow across the range of soils, initial and boundary conditions found in practice

Water flow in partially saturated porous media is of importance in many disciplines such as hydrology, agriculture, environmental management or geotechnical engineering. A robust and accurate solution methodology applicable across the range of soils, initial and boundary conditions found in practice is difficult to identify. Three mass conservative iteration schemes for the approximation of the flow equation in partially saturated porous media are evaluated considering large contrasts in soil texture, initial moisture and boundary conditions typically found in real world application. The three schemes reviewed are the well established modified Picard algorithm (Celia-Scheme), and two promising approaches the L-Scheme and the Casulli-Scheme. A systematic variation of soil texture and initial moisture in an intense rainfall infiltration scenario is set up in order to consider the numerical challenges associated with high non-linearity. The present analysis provides insights into robustness and efficiency in solving variably saturated flow across the range of soils, initial and boundary conditions found in practice. Casulli’s scheme, in particular, provides an acceptable compromise between robustness and execution duration even under the severest conditions investigated, while the other schemes sacrifice one for the sake of the other. The authors believe that advances in the development of numerical schemes for partly saturated flow require the kind of methodology presented here in combination with relevant benchmark problems in order to enable a fair evaluation and comparison of numerical techniques across the related disciplines.

Spectral deferred correction method for Landau–Brazovskii model with convex splitting technique

The Landau–Brazovskii model is a well-known Landau model for finding the complex phase structures in microphase-separating systems ranging from block copolymers to liquid crystals. It is critical to design efficient numerical schemes for the Landau–Brazovskii model with energy dissipation and mass conservation properties. Here, we propose a mass conservative and energy stable scheme by combining the spectral deferred correction (SDC) method with the convex splitting technique to solve the Landau–Brazovskii model efficiently. An adaptive correction strategy for the SDC method is implemented to reduce the cost time and preserve energy stability. Numerical experiments, including two- and three-dimensional periodic crystals in Landau–Brazovskii model, are presented to show the efficiency of the proposed numerical method.

A Strongly Mass Conservative Method for the Coupled Brinkman-Darcy Flow and Transport

In this paper, a strongly mass conservative and stabilizer-free scheme is designed and analyzed for the coupled Brinkman–Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered discontinuous Galerkin scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method.

A mass conservative, well balanced and positivity-preserving central scheme for shallow water equations

Based on the invariant-region-preserving (IRP) reconstruction method introduced in [Yan, Tong and Chen, Appl. Math. Comput., 436 (2023) 127500], a second order unstaggered central scheme is proposed to solve the shallow water equations with bottom topography in the framework that the bottom is discretized by a continuous, piecewise linear approximation. The reconstruction applies a modification locally on a preliminary reconstructed surface gradient in every cell to yield a convexity property of the sampled point value in the forward and backward projections. The water mass conservation is proved by rewriting the scheme in a conservation form. The modification does not change the preliminary reconstructed slope of water surface for the lake-at-rest steady state and then keeps the well-balancing property of the surface gradient method. The convexity property ensures the nonnegativity of the updated water depth under a large CFL number which yields a considerable speed-up. The numerical experiments are shown to demonstrate the robustness of the scheme.

Conservative compact difference scheme for the generalized Korteweg-de Vries equation

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

Large temperature difference heat dominated flow simulations using a pressure-based lattice Boltzmann method with mass correction

This paper addresses simulation of heat dominated compressible flows in a closed cavity using a pressure-based lattice Boltzmann (LB) method, in which thermal effects are modeled by applying a pressure-featured zero-order moment of distribution functions. A focus is made on the conservation of mass at boundary nodes, which is a challenging issue that significantly complicated by the density-decoupled zero-order moment here. The mass leakage at boundary nodes is mathematically quantified, which enables an efficient local mass correction scheme. The performance of this solver is assessed by simulating buoyancy-driven flows in a closed deferentially heated cavity with large temperature differences (non-Boussinesq) at Rayleigh numbers ranging from 103 to 107. Simulations show that mass leakage at solid walls in such configurations is a critical issue to obtain reliable solutions, and it eventually leads to simulations overflow when the cavity is inclined. The proposed mass correction scheme is, however, shown to be effective to control the mass leakage and get accurate solutions. Thus, associated with the proposed mass conservation scheme, the pressure-based LB method becomes reliable to study natural convection dominated flows at large temperature differences in closed geometries with mesh aligned boundaries or not.

A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation

The primary challenge in solving kinetic equations, such as the Vlasov equation, is the high-dimensional phase space. In this context, dynamical low-rank approximations have emerged as a promising way to reduce the high computational cost imposed by such problems. However, a major disadvantage of this approach is that the physical structure of the underlying problem is not preserved. In this paper, we propose a dynamical low-rank algorithm that conserves mass, momentum, and energy as well as the corresponding continuity equations. We also show how this approach can be combined with a conservative time and space discretization.

Mass-conserving diffusion-based dynamics on graphs

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

Practical Application of the Galerkin Finite Element Method with a Mass Conservation Scheme under Dirichlet Boundary Conditions to Solve Groundwater Problems

The Galerkin finite element method (FEM) has long been used to solve groundwater flow equations and compute the mass balance in a region. In this study, we proposed a simple, new computational FEM procedure for global mass balance computations that can simultaneously obtain boundary fluxes at Dirichlet boundary nodes and finite element hydraulic heads at all nodes in only one step, whereas previous approaches usually require two steps. In previous approaches, the first step obtains the Galerkin finite element hydraulic heads at all nodes, and then, the boundary fluxes are calculated using the obtained Galerkin finite element hydraulic heads in a second step. Comparisons between the new approach proposed in this study and previous approaches, such as Yeh’s approach and a conventional differential approach, were performed using two practical groundwater problems to illustrate the improved accuracy and efficiency of the new approach when computing the global mass balance or boundary fluxes. From the results of the numerical experiments, it can be concluded that the new approach provides a more efficient mass balance computation scheme and a much more accurate mass balance computation compared to previous approaches that have been widely used in commercial and public groundwater software.

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM, 42(3):934–952, 2004).

A New Simulation Framework for Soil–Root Interaction, Evaporation, Root Growth, and Solute Transport

Core Ideas We present a locally mass‐conservative flow, transport, and root–soil interaction model. We include a sustainable, flexible research software framework for plant‐scale model development. This is an improved model concept for fluid dynamic processes, evaporation, root water uptake. We have developed a general model concept and a flexible software framework for the description of plant‐scale soil–root interaction processes including the essential fluid mechanical processes in the vadose zone. The model was developed in the framework of non‐isothermal, multiphase, multicomponent flow and transport in porous media. The software is an extension of the open‐source porous media flow and transport simulator DuMux to embedded mixed‐dimensional coupled schemes. Our coupling concept allows us to describe all processes in a strongly coupled form and adapt the complexity of the governing equations in favor of either accuracy or computational efficiency. We have developed the necessary numerical tools to solve the strongly coupled nonlinear partial differential equation systems that arise with a locally mass conservative numerical scheme even in the context of evolving root architectures. We demonstrate the model concept and its features, discussing a virtual hydraulic lift experiment including evaporation, root tracer uptake on a locally refined grid, the simultaneous simulation of root growth and root water uptake, and an irrigation scenario comparing different models for flow in unsaturated soil. We have analyzed the impact of evaporation from soil on the soil water distribution around a single plant's root system. Moreover, we have shown that locally refined grids around the root system increase computational efficiency while maintaining accuracy. Finally, we demonstrate that the assumptions behind the Richards equation may be violated under certain conditions.

A Mass Conservative Scheme for Fluid–Structure Interaction Problems by the Staggered Discontinuous Galerkin Method

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method.

A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities

NWO support through the Visitors Grant (040.11.351 to J.M.N.); Meltzer foundation, University of Bergen and the NWO Visitors Grant (040.11.499 to F.A.R.); Research Foundation–Flanders FWO through the Odysseus programme (G0G1316N to I.S.P.).

A mass conservative scheme for solving the Vlasov–Poisson equation using characteristic curve

In this paper, we introduce a mass conservative scheme for solving the Vlasov–Poisson equation. This scheme is based on an Eulerian approach and is constructed using an interpolation scheme with limiters. In order to preserve the mass, the difference in the values for numerical flux functions on each cell is used; for this, the flux functions are constructed by preserving both the solution along a characteristics and the mass in each cell. We mainly investigate the conservation of L 1 and L 2 norms of the distribution function, total energy, entropy, and minimum value. In addition, we show that this scheme is bounded on the total variation. To demonstrate the efficiency of the proposed scheme, this scheme is compared with the flux balance scheme, Positive and Flux Conservative scheme, Umeda’s scheme, and fifth order WENO reconstruction finite volume scheme.

Development of the Eulerian atmospheric transport model GEARN-FDM: Validation against the European tracer experiment

In this study, we developed an atmospheric dispersion model based on the finite difference method (GEARN-FDM) to simulate long-range radiological dispersion. The model includes a mass conservation and monotonic advection scheme and a horizontal diffusion scheme based on the deformation of the resolved horizontal wind. Meteorological fields in calculating dispersion were simulated by the Advanced Research Weather Research and Forecasting (WRF) model. ERA-interim reanalysis was used for the WRF nudging calculation. By using tracer gas concentrations from the European tracer experiment, the GEARN-FDM performance could be tested. The results indicated high performance with factors of 2 and 5 of 37% and 72%, respectively. To investigate the impact of horizontal diffusivity on the concentration distribution, we examined an additional simulation run using GEARN-FDM with a constant value of horizontal diffusivity. In comparison with the additional run, the plume calculated in the original GEARN-FDM run was locally and temporally diluted due to large horizontal diffusivity in a mountainous region. Therefore, geographical heterogeneity of horizontal diffusivity may affect regional-scale atmospheric dispersion simulations.

A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

ABSTRACTThis paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

Parallel implementation of the cascade mass-conserving semi-Lagrangian transport scheme

AbstractModern atmospheric models for climate simulations require accurate and efficient, locally mass-conservative and monotonic numerical schemes for treating the transport of atmospheric constituents. One of the ways to design such schemes is Finite-Volume Semi-Lagrangian approach (FVSL). FVSL schemes are characterised by the computational efficiency advantage due to the possibility of using large time-steps and efficient treatment of multiple transported quantities (tracers). This article presents massively parallel and multi-tracer efficient version of the recently developed 3D cascade FVSL transport scheme. Using hybrid distributed-shared memory parallelism with 1D MPI domain decomposition in latitude and OpenMP computations for longitude loops allows to use efficiently up to 1600 computational cores. We hope this number will grow with the future growth of the number of shared memory cores per computational node. Multi-tracer optimisations of the scheme (mostly, developing multi-tracer efficient monotonic filter) allow to reduce the cost of transporting additional tracer to 18–23% of running the scheme with one tracer.

A three-dimensional volume-of-fluid method for reconstructing and advecting three-material interfaces forming contact lines

We introduce a piecewise-linear, volume-of-fluid method for reconstructing and advecting three-dimensional interfaces and contact lines formed by three materials. The new method employs a set of geometric constructs that can be used in conjunction with any volume-tracking scheme. In this work, we used the mass-conserving scheme of Youngs to handle two-material cells, perform interface reconstruction in three-material cells, and resolve the contact line. The only information required by the method is the available volume fraction field. Although the proposed method is order dependent and requires a priori information on material ordering, it is suitable for typical contact line applications, where the material representing the contact surface is always known. Following the reconstruction of the contact surface, to compute the interface orientation in a three-material cell, the proposed method minimizes an error function that is based on volume fraction distribution around that cell. As an option, the minimization procedure also allows the user to impose a contact angle. Performance of the proposed method is assessed via both static and advection test cases. The tests show that the new method preserves the accuracy and mass-conserving property of the Youngs method in volume-tracking three materials.