Discrete Continuity Equation Research Articles

Artificial patterns in spatially discrete models of cell migration and how to mitigate them

Several discrete models for diffusive motion are known to exhibit checkerboard artifacts, absent in their continuous analogues. We study the origins of the checkerboard artifact in the discrete heat equation and show that this artifact decays exponentially in time when following either of two strategies: considering the present state of each lattice site to determine its own future state (self-contributions), or using non-square lattice geometries. Afterwards, we examine the effects of these strategies on nonlinear models of biological cell migration with two kinds of cell-cell interactions: adhesive and polar velocity alignment. In the case of adhesive interaction, we show that growing modes related to pattern formation overshadow artifacts in the long run; nonetheless, artifacts can still be completely prevented following the same strategies as in the discrete heat equation. On the other hand, for polar velocity alignment we show that artifacts are not only strengthened, but also that new artifacts can arise in this model which were not observed in the previous models. We find that the lattice geometry strategy works well to alleviate artifacts. However, the self-contribution strategy must be applied more carefully: lattice sites should contribute to both their own density and velocity values, and their own velocity contribution should be high enough. With this work, we show that these two strategies are effective for preventing artifacts in spatial models based on the discrete continuity equation.

A mass-conservative semi-implicit volume of fluid method for the Navier–Stokes equations with high order semi-Lagrangian advection scheme

This paper deals with the development of a semi-implicit numerical method for the Navier–Stokes equations using the non-linear volumes of fluid (VOF) approach and a semi-Lagrangian scheme for the discretization of the advection contribution based on a high order reconstruction of the velocity field. The VOF approach guarantees high flexibility and is able to reproduce several phenomena that appear in real scenarios such as free surface flows, pressurized channels and jets. The discrete velocity field from the momentum conservation law is formally inserted into the discrete continuity equation, hence yielding a mildly non-linear system for the unknown hydraulic head which can be solved through a nested Newton-type algorithm. The computation of the non-linear convective diffusion contribution is then based on a high order reconstruction of the velocity field, which is furthermore constrained to exactly recover the original pointwise values of the numerical solution. As a consequence, the mass conservation is fully preserved while providing information about the main velocity field and its high order moments, later employed in the computation of the Lagrangian trajectories needed for the discretization of the convective and diffusive terms. Furthermore, the bottom friction and the tangential stresses can be directly computed from the high order velocity reconstruction. The method is derived in a general form with the only requirement to be structured in the z− direction, so that it applies to the three- and the two-dimensional cases with unstructured grids in the horizontal space. Convergence studies are carried out to demonstrate the accuracy of the reconstruction operator. Finally, the numerical scheme is validated against several benchmarks that include 2Dxz, 2Dxy and 3D non-hydrostatic flows with complex geometry in order to show the flexibility of the proposed algorithm, including a real-world application.

An approach to accelerate the convergence of SIMPLER algorithm for convection-diffusion problems of fluid flow with heat transfer and phase change

A new segregated pressure-based algorithm is developed to solve two-dimensional incompressible fluid flows with convective heat transfer and liquid to solid phase change by the finite volume method. The predictive-correction scheme is based on the SIMPLER algorithm with n added iterative cycles to increase the accuracy of pressure in satisfying the discretized continuity equation. The performance of SIMPLERnP is compared with SIMPLER and IDEAL in the solution of the laminar fluid flow in a lid driven cavity, natural heat convection inside a square cavity, convective heat transfer in the annular space between two concentric cylinders and conjugate natural heat convective solidification of a binary alloy that included the transient heat conduction in the thick mold walls. The results indicate that the proposed algorithm is highly stable, even when a value of 0.99 is used for the under-relaxation of the velocity components, and allows significant improvements in the reduction of the number of iterations and time of computation.

Multiphase smoothed particle hydrodynamics modeling of forced liquid sloshing

AbstractIn the present paper, an improved multiphase weakly compressible smoothed particle hydrodynamics model for balancing the accuracy and stability of the long‐term simulations is proposed to model the forced liquid sloshing in a tank. The governing equations of the multiphase flow are discretized by considering the density discontinuity over the interface. To suppress the pressure oscillation, a previous density correction term suitable only for single‐phase problems is modified and incorporated into the discrete continuity equation to suit multiphase problems. The modified density reinitialization algorithm is implemented to calculate the pressure of the boundary particles, and the coupled dynamic solid boundary treatment (SBT) is employed to determine the rigid wall condition. For convenience, a numerical probe algorithm is also proposed to accurately measure the wave height. The present model exhibits a better numerical stability than the previous multiphase smoothed particle hydrodynamics model, and its results well confirm with the experimental data of the forced sloshing of liquid excited by swaying or rolling.

An integrated smoothed particle hydrodynamics model for complex interfacial flows with large density ratios

AbstractIn this paper, an integrated smoothed particle hydrodynamics (SPH) model for complex interfacial flows with large density ratios is developed. The discrete continuity equation and acceleration equation are obtained by considering the time derivative of the volume of particle and Eckart's continuum Lagrangian equation. A continuum surface force model is used to meet the fact that surface force may not be distributed uniformly on each side of the interface. An improved boundary condition is imposed to model wall free‐slip and no‐slip condition for interfacial flows with large density ratios. Particle shifting algorithm (PSA) is added for interfacial flows by imposing the normal correction near the interface, called as Interface‐PSA. Then four representative numerical examples, including droplet deformation, Rayleigh‐Taylor instability, dam breaking, and bubble rising, are presented and compared well with reference data. It is demonstrated that inherent interfacial flow physics can be well captured, including surface tension and the dynamic evolution of the complex interfaces.

A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous Galerkin schemes for the shallow water equations

We propose a novel family of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin (DG) schemes on staggered Cartesian grids for the solution of the shallow water equations expressed in conservative form in one and two space dimensions.We start from the unlimited arbitrary high order accurate staggered semi-implicit DG scheme proposed by Dumbser and Casulli (2013). In this method, the continuity equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. In the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block penta-diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. However, according to Godunov's theorem, any unlimited high order scheme inevitably produces spurious oscillations in the vicinity of discontinuities and strong gradients.In the present paper, we therefore propose to extend the successful family of a posteriori subcell finite volume limiters recently introduced by Dumbser et al. (2014) for explicit DG schemes also to semi-implicit time discretizations. At time tn the unlimited DG scheme is run in order to produce a so-called candidate solution for time tn+1. Then, the cells characterized by a non-admissible candidate solution are found by using physical and numerical detection criteria based on the positivity of the solution, the absence of floating point errors and the use of a relaxed discrete maximum principle (DMP) according to the MOOD strategy of Clain, Loubère and Diot (2013). In all the cells that are flagged as troubled control volumes a more robust semi-implicit finite volume (FV) method is then applied on a sub-grid composed of 2P+1 cells, where P denotes the polynomial degree used for approximating the discrete solution within the DG scheme. Then, after having identified the troubled cells, the linear system for the new free surface elevation is assembled and solved again, where unlimited cells use the high order semi-implicit DG scheme and limited cells are evolved via the more robust finite volume method. Finally, from the subcell finite volume averages a higher order DG polynomial is reconstructed and the scheme proceeds with the next time step.We apply the new semi-implicit staggered DG method with a posteriori subcell FV limiter to classical benchmarks such as Riemann problems in 1D and circular dam-break problems in 2D with shock waves, showing that the new subcell finite volume limiter is able to resolve shocks accurately without producing spurious oscillations. Moreover, if the solution is smooth, the detector does not find any troubled cells, as expected; consequently, the limiter is not activated and the method corresponds to the unlimited staggered semi-implicit DG scheme. In addition, we carry out numerical tests which show that the new scheme is well-balanced and able to deal with wet and dry fronts.

A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier–Stokes equations

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier–Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H(div) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.

Solution of lid-driven cavity problems with an improved SIMPLE algorithm at high Reynolds numbers

The SIMPLE algorithm and its variants have been successfully employed to solve the driven-cavity problem at Re<10,000. However, converged steady solutions cannot be achieved at higher Re (15,000–30,000) by SIMPLE-family algorithms owing to its inherent drawbacks. In this work, a new segregated solver, ISTEC-N (incompressible flow solver with two explicit N correction loops), for determining the solution of incompressible flow on structured collocated grid systems is proposed. A pressure equation is deduced from the discretized continuity equation to overcome the “Semi-Implicit” characteristic of the SIMPLE algorithm. An explicit velocity update loop is introduced to improve the consistency between the pressure field and the velocity field in each outer iteration. The performance of the ISTEC-N algorithm is evaluated by solving the driven-cavity problem at high Re number of (15,000<Re<30,000). It is shown that the ISTEC-N algorithm manifests a strong robustness feature under these high convection conditions. The computational results given by the ISTEC-N algorithm are in excellent agreement with the benchmarking data.

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier–Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Although the resulting adaptive staggered grids are more complex than classical uniform Cartesian meshes, the numerical scheme can still be written in a rather compact form. Thanks to the use of a tensor-product formulation for the definition of the nodal basis in the d-dimensional space (d=2,3), all the discrete operators can be efficiently written as a combination of linear one-dimensional operators acting in the d space directions separately.Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier–Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. In order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive terms in the momentum equation. A particular feature of our staggered DG approach is that the viscous stress tensor is discretized on the dual mesh. This corresponds to the use of a lifting operator, but on the staggered grid. Both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fast convergence is observed. Moreover, it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner. Due to the explicit discretization of the nonlinear convective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicit DG methods based on the fluid velocity. Moreover, by using a suitable local time stepping technique based on the ADER-DG framework, each element can use its optimal local time step also within the AMR context.The new space-time adaptive staggered DG scheme has been thoroughly verified for polynomial degrees up to N=9 for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. The high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution.To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier–Stokes equations on space-time adaptive meshes in two and three space dimensions.

Family of Runge–Kutta-Based Algorithms for Unsteady Incompressible Flows

A family of algorithms based on implicit Runge–Kutta time integration suitable for unsteady incompressible flow is developed. Spatially discretized momentum equations are integrated in time using implicit Runge–Kutta methods. An equation for pressure is formed by combining the discretized continuity equation with the discretized momentum equations. This family of algorithms, called IRK-SIMPLER, uses only exact discretized mass and momentum equations and requires no relaxation to converge. Two distinct IRK-SIMPLER variants are analyzed. The first variant solves the pressure equation once per time step and has the advantage of fewer computations per time step but is limited to first-order accuracy in time. The second variant centers on reformulating a pressure equation each Runge–Kutta stage by manipulating the momentum stage equations, and this pressure equation is solved multiple times within a time step. The second variant is shown to achieve a higher temporal order of accuracy. By analyzing the largest time step allowable and run time required to achieve time-accurate solutions, the most efficient algorithms in this family are delineated. For the chosen problems, IRK-SIMPLER is able to achieve as high as 101 times speedup compared to the traditional SIMPLER algorithm with Crank–Nicolson time integration using a finite volume discretization scheme.

Applications of parallel computing technologies for modeling the mixed convection in backward-facing step flows with the vertical buoyancy forces

This paper presents numerical solutions of the mixed convection in backward-facing step flows with the vertical buoyancy forces. A two-dimensional incompressible Navier-Stokes equation is used to describe this process. This system is approximated by the control volume method and solved numerically by the projection method. The two-dimensional Poisson equation satisfying the discrete continuity equation that is solved by the Jacobi iterative method at each time step. The numerical solutions of the laminar flow behind the backward-facing step with the vertical buoyancy forces are compared with the numerical results of other authors. This numerical algorithm is completely parallelized using various geometric domain decompositions (1D, 2D and 3D). Preliminary theoretical analysis of the various decomposition methods effectiveness of the computational domain and real computational experiments for this problem were made and the best domain decomposition method was determined. In the future, a proven mathematical model and parallelized numerical algorithm with the best domain decomposition method can be applied for various complex flows with the vertical buoyancy forces. G M T Detect language Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Text-to-speech function is limited to 200 characters Options : History : Feedback : Donate Close

Elementary dispersion analysis of some mimetic discretizations on triangular C-grids

Spurious modes supported by triangular C-grids limit their application for modeling large-scale atmospheric and oceanic flows. Their behavior can be modified within a mimetic approach that generalizes the scalar product underlying the triangular C-grid discretization. The mimetic approach provides a discrete continuity equation which operates on an averaged combination of normal edge velocities instead of normal edge velocities proper. An elementary analysis of the wave dispersion of the new discretization for Poincaré, Rossby and Kelvin waves shows that, although spurious Poincaré modes are preserved, their frequency tends to zero in the limit of small wavenumbers, which removes the divergence noise in this limit. However, the frequencies of spurious and physical modes become close on shorter scales indicating that spurious modes can be excited unless high-frequency short-scale motions are effectively filtered in numerical codes. We argue that filtering by viscous dissipation is more efficient in the mimetic approach than in the standard C-grid discretization. Lumping of mass matrices appearing with the velocity time derivative in the mimetic discretization only slightly reduces the accuracy of the wave dispersion and can be used in practice. Thus, the mimetic approach cures some difficulties of the traditional triangular C-grid discretization but may still need appropriately tuned viscosity to filter small scales and high frequencies in solutions of full primitive equations when these are excited by nonlinear dynamics.

A staggered space–time discontinuous Galerkin method for the three-dimensional incompressible Navier–Stokes equations on unstructured tetrahedral meshes

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the three-dimensional incompressible Navier–Stokes equations on staggered unstructured curved tetrahedral meshes. As is typical for space–time DG schemes, the discrete solution is represented in terms of space–time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier–Stokes equations. Similarly to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. While staggered meshes are state of the art in classical finite difference schemes for the incompressible Navier–Stokes equations, their use in high order DG schemes is still quite rare. A very simple and efficient Picard iteration is used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner, even up to very high polynomial degrees. For a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point, the resulting system is, in addition, symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method.The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data.To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the three-dimensional incompressible Navier–Stokes equations on staggered unstructured tetrahedral grids.

A hybrid finite-volume/finite-difference-based one-dimensional Boussinesq model for waves attenuated by vegetation

A hybrid finite-volume/finite-difference scheme is proposed to solve the one-dimensional Boussinesq equations for wave attenuation by vegetation. The effect of vegetation is included as a source term in a form of drag force. The convective part of the equations is discretized by the finite-volume method, while the finite-difference method is used to discretize the remaining terms. The variable values for the local Riemann problem at each cell face are calculated by a fourth-order MUSCL reconstruction method. The source terms and the dispersion terms are discretized using the centered finite-difference schemes up to fourth-order accuracy. The unsteady terms are discretized by the second-order MUSCL-Hancock scheme. The discretized continuity equation is solved explicitly, while the discretized momentum equation is solved using the Thomas algorithm. The developed Boussinesq model is tested with analytical solutions and reported experimental data. To further validate the model, the computed results are compared with the experimental data observed in two vegetated wave flumes. It is demonstrated that the developed model is suitable for predicting wave propagation in vegetated water bodies.

Handling the divergence constraints in Maxwell and Vlasov–Maxwell simulations

The aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the Vlasov–Maxwell equations and the Maxwell equations with given sources. These methods can be classified in two types: field correction methods and source correction methods. The field correction methods introduce new unknowns in the equations, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss’ law that needs to be defined in accordance with the discretization of the Maxwell propagation operator.

A staggered space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations on two-dimensional triangular meshes

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier–Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. Note that the same space–time DG scheme on a collocated grid would lead to ten non-zero blocks per element, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier–Stokes equations. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is verified for approximation polynomials of degree up to p=4 in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the incompressible Navier–Stokes equations on staggered unstructured grids.

Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media

Numerical simulation of multiphase flow in porous media is of great importance in a wide range of applications in science and engineering. The governing equations are the continuity equation and Darcy’s law. A novel control volume finite element (CVFE) approach is developed to discretize the governing equations in which a node-centered control volume approach is applied for the saturation equation, while a CVFE method is used for discretization of the pressure equation. We embed the discrete continuity equation into the pressure equation and ensure that the continuity equation is exactly enforced. Furthermore, the scheme is equipped with dynamic anisotropic mesh adaptivity which uses a metric tensor field approach, based on the curvature of fields of interest, to control the size and shape of elements in the metric space. This improves the resolution of the mesh in the zones of dynamic interest. Moreover, the mesh adaptivity algorithm employs multi-constraints on element size in different regions of the porous medium to resolve multi-scale transport phenomena. The advantages of mesh adaptivity and the capability of the scheme are demonstrated for simulation of flow in several challenging computational domains. The scheme captures the key features of flow while preserving the initial geometry and can be applied for efficient simulation of flow in heterogeneous porous media and geological formations.

A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations

In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discrete velocity vector field is defined on an edge-based dual grid. The flexibility of high order DG methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids.Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non-zero blocks per element for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to p=3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data.

Development of a method for calculating high-dimension heat networks

A transition from the mass balance equations based on Kirchhoff’s first and second laws to modeling on the basis of a discretized continuity equation is made for describing a hydraulic network. A technique for calculating high-dimension hydraulic and heat networks based on the numerical finite-difference control volume method is developed. Unlike the existing approaches, the proposed technique does not involve the need to determine hydraulic loops and boils down to solving the problem of obtaining a unified field of pressures for the entire calculation region. This advantage of the proposed method opens the possibility of applying it for solving high-dimension problems containing more than a million of hydraulic links. The proposed numerical method features stable operation for hydraulic networks the neighboring links of which may have pressure drop coefficients differing from each other by more than 10 orders of magnitude. In contrast, the global gradient algorithm implemented in the standard software system EPANET is of little use for such applications. The convergence rate of the proposed technique is close to that of the Todini gradient algorithm and is almost independent of the problem dimension.

Charge-conserving grid based methods for the Vlasov–Maxwell equations

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver.