On the hyperbolicity of certain models of polydisperse sedimentation

The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first‐order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Bürger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the models by Batchelor and Wen, Höfler and Schwarzer, and Davis and Gecol, for which the Jacobian of the flux is a rank‐3 or rank‐4 perturbation of a diagonal matrix. Explicit estimates of the regions of hyperbolicity of these models are derived via the approach of the so‐called secular equation (J. Anderson. Linear Algebra and Applications 1996; 246:49–70), which identifies the eigenvalues of the Jacobian with the poles of a particular rational function. Hyperbolicity of the system is guaranteed if the coefficients of this function have the same sign. Sufficient conditions for this condition to be satisfied are established for each of the models considered. Some numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

On mathematical models and numerical simulation of the fluidization of polydisperse suspensions

The well-known Masliyah–Lockett–Bassoon (MLB) model for sedimentation of small particles is extended to fluidization of polydisperse suspensions. For N particle species that differ in size and density, this model leads to a first-order system of N conservation laws, which in general is of mixed (in the case N = 2, hyperbolic–elliptic) type. By a simple algebraic steady-state analysis, we derive necessary compatibility conditions on the size and density parameters that admit the formation of stationary fluidized beds. We then proceed to determine the composition of polydisperse fluidized beds of given compatible species by varying the fluidization velocity and the initial composition of the suspensions, and prove that, within the framework of the MLB model combined with the Richardson–Zaki formula, the constructed bidisperse beds always cause the equations to be hyperbolic. This means that these states are always predicted to be stable. The transient behaviour of the MLB model applied to fluidization is illustrated by three numerical examples, in which the system of conservation laws is solved for N = 2, N = 3 and N = 5, respectively. These examples illustrate the effects of bed expansion and layer inversion caused by successively increasing the applied fluidization velocity and show that the predicted fluidized states are indeed attained.     

Conservative semi‐Lagrangian finite volume schemes

Semi‐Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area‐weighting. Numerical results are presented illustrating some of the features of these schemes. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:403–425, 2001     

A preliminary investigation on an ELLAM scheme for linear transport equations

We present an Eulerian‐Lagrangian localized adjoint method (ELLAM) for linear advection‐reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to investigate the performance of the ELLAM scheme with a range of well‐perceived and widely used methods in fluid dynamics including the monotonic upstream‐centered scheme for conservation laws (MUSCL), the minmod method, the flux‐corrected transport method (FCT), and the essentially non‐oscillatory (ENO) schemes and weighted essentially non‐oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is very competitive with these methods in the context of linear transport PDEs, and suggest/justify the development of ELLAM‐based simulators for subsurface porous medium flows and other applications. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 22–43, 2003     

On second-order antidiffusive Lagrangian-remap schemes for multispecies kinematic flow models

This paper focuses on the numerical approximation of the solutions of multi-species kinematic flow models. These models are strongly coupled nonlinear first-order conservation laws with various applications like sedimentation of a polydisperse suspension in a viscous fluid, or traffic flow modeling. Since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form, this is a challenging issue. A new class of simple schemes based on a Lagrangian- Eulerian decomposition (the so-called Lagrangian-remap (LR) schemes) was recently advanced in [4] for traffic flow models with nonnegative velocities, and extended to models of polydisperse sedimentation in [5]. These schemes are supported by a partial numerical analysis when one species is considered only, and turned out to be competitive in both accuracy and efficiency with several existing schemes. Since they are only first-order accurate, it is the purpose of this contribution to propose an extension to second-order accuracy using quite standard MUSCL and Runge-Kutta techniques. Numerical illustrations are proposed for both applications and involving eleven species (sedimentation) and nine species (traffic) respectively.     

An Eulerian–Lagrangian Weighted Essentially Nonoscillatory scheme for nonlinear conservation laws

We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL‐WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi‐Lagrangian) scheme using a fixed velocity v. A flux correction computation accounts for particles that cross the v‐traceline during the time step. If v = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when v is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps. The v‐traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re‐averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 651–680, 2017     

Symmetry analysis of the charged squashed Kaluza–Klein black hole metric

In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space‐time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space‐time metric using Lie point symmetries is presented, and then the n‐dimensional optimal system of this space‐time metric, for n = 1,…,4, are computed. It is shown that there is no any n‐dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for n≥5. Then the point symmetries of the one‐parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017     

Non‐standard finite difference schemes for multi‐dimensional second‐order systems in non‐smooth mechanics

This work is an extension of the paper (Proc. R. Soc. London 2005; 461A :1927–1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these non‐smooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non‐standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Some techniques for improving the resolution of finite difference component-wise WENO schemes for polydisperse sedimentation models

Polydisperse sedimentation models can be described by a system of conservation laws for the concentration of each species of solids. Some of these models, as the Masliyah–Locket–Bassoon model, can be proven to be hyperbolic, but its full characteristic structure cannot be computed in closed form. Component-wise finite difference WENO schemes may be used in these cases, but these schemes suffer from an excessive diffusion and may present spurious oscillations near shocks. In this work we propose to use a flux-splitting that prescribes less numerical viscosity for component-wise finite difference WENO schemes. We compare this technique with others to alleviate the diffusion and oscillatory behavior of the solutions obtained with component-wise finite difference WENO methods.     

Efficient local structure‐preserving schemes for the RLW‐type equation

The multisymplectic schemes have been used in numerical simulations for the RLW‐type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure‐preserving schemes for the RLW‐type equation, which preserve the local energy exactly on any time‐space region and can produce richer information of the original problem. The schemes will be mass‐ and energy‐preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant‐preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678–1691, 2017     

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high‐order quasi‐compact schemes for space fractional diffusion equations. Because of the quasi‐compactness of the derived schemes, no points beyond the domain are used for all the high‐order schemes including second‐order, third‐order, fourth‐order, and even higher‐order schemes; moreover, the algebraic equations for all the high‐order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the fractional derivatives with nonhomogeneous boundaries are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1345–1381, 2015     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

Continuum shock profiles for discrete conservation laws II: Stability

We show that the continuum shock profiles for dissipative difference schemes constructed in Part I are nonlinearly stable. It is shown first that the profiles have the conservation property, obtained as the limit of the discrete version for profiles with nearby rational, quasi‐Diophantine speeds. This allows us to formulate antidifferencing of the schemes and to apply a generalization of the pointwise approach for viscous conservation laws for the stability analysis. © 1999 John Wiley & Sons, Inc.     

A 2D Staggered Multi-Material ALE Code Using MOF Interface Reconstruction

Hydrocodes are necessary numerical tools in the fields of implosion and high-velocity impact, which often involve large deformations with changing-topology interfaces. It is very difficult for Lagrangian or Simplified Arbitrary Lagrangian-Eulerian (SALE) codes to tackle these kinds of large-deformation problems, so a staggered Multi-Material ALE (MMALE) code is developed in this paper, which is the explicit time-marching Lagrange plus remap type. We use the Moment Of Fluid (MOF) method to reconstruct the interfaces of multi-material cells and present an adaptive bisection method to search for the global minimum value of the nonlinear objective function. To keep the Lagrangian computations as long as possible, we develop a robust rezoning method named as Combined Rezoning Method (CRM) to generate the convex, smooth grids for the large-deformation domain. Regarding the staggered remap phase, we use two methods to remap the variables of Lagrangian mesh to the rezoned one. One is the first-order intersection-based remapping method that doesn't limit the distances between the rezoned and Lagrangian meshes, so it can be used in the applications of wide scope. The other one is the conservative second-order flux-based remapping method developed by Kucharika and Shashkov [22] that requires the rezoned element to locate in its adjacent old elements. Numerical results of triple point problem show that the result of first-order remapping method using ALE computations is gradually convergent to that of second-order remapping method using Eulerian computations with the decrease of rezoning, thereby telling us that MMALE computations should be performed as few as possible to reduce the errors of the interface reconstruction and the remapping. Numerical results provide a clear evidence of the robustness and the accuracy of this MMALE scheme, and that our MMALE code is powerful for the large-deformation problems.     

A combinatorial approach to transitive extensions of generously unitransitive permutation groups

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H, Ω) on the k‐subsets of Ω, k=2, 3, 4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer‐generated data for the particular series of Hamming schemes H(m, 3), m⩾2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:369–391, 2010     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

On Moving Frames and Noether’s Conservation Laws

Noether’s Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler‐Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. For the examples, we demonstrate, knowledge of this structure allows the Euler‐Lagrange equations to be integrated with relative ease. Our methods take advantage of recent advances in the theory of moving frames by Fels and Olver, and in the symbolic invariant calculus by Hubert. The results here generalize those appearing in Kogan and Olver [ 1 ] and in Mansfield [ 2 ]. In particular, we show results for high‐dimensional problems and classify those for the three inequivalent SL(2) actions in the plane.     

Analysis of the one-dimensional Euler–Lagrange equation of continuum mechanics with a Lagrangian of a special form

The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed.One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.