<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

Lattice Boltzmann modeling of pore‐scale fluid flow through idealized porous media

In order to capture the hydro‐mechanical impacts on the solid skeleton imposed by the fluid flowing through porous media at the pore‐scale, the flow in the pore space has to be modeled at a resolution finer than the pores, and the no‐slip condition needs to be enforced at the grain–fluid interface. In this paper, the lattice Boltzmann method (LBM), a mesoscopic Navier–Stokes solver, is shown to be an appropriate pore‐scale fluid flow model. The accuracy and lattice sensitivity of LBM as a fluid dynamics solver is demonstrated in the Poiseuille channel flow problem (2‐D) and duct flow problem (3‐D). Well‐studied problems of fluid creeping through idealized 2‐D and 3‐D porous media (J. Fluid Mech. 1959; 5 (2):317–328, J. Fluid Mech. 1982; 115 :13–26, Int. J. Multiphase Flow 1982; 8 (4):343–360, Phys. Fluids A 1989; 1 (1):38–46, Int. J. Numer. Anal. Meth. Geomech. 1999; 23 :881–904, Int. J. Numer. Anal. Meth. Geomech. 2010; DOI: 10.1002/nag.898, Int. J. Multiphase Flow 1982; 8 (3):193–206) are then simulated using LBM to measure the friction coefficient for various pore throats. The simulation results agree well with the data reported in the literature. The lattice sensitivity of the frictional coefficient is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.     

A solution adaptive simulation of compressible multi‐fluid flows with general equation of state

The unstructured quadrilateral mesh‐based solution adaptive method is proposed in this article for simulation of compressible multi‐fluid flows with a general form of equation of state (EOS). The five equation model (J. Comput. Phys. 2002; 118 :577–616) is employed to describe the compressible multi‐fluid flows. To preserve the oscillation‐free property of velocity and pressure across the interface, the non‐conservative transport equation is discretized in a compatible way of the HLLC scheme for the conservative Euler equations on the unstructured quadrilateral cell‐based adaptive mesh. Five numerical examples, including an interface translation problem, a shock tube problem with two fluids, a solid impact problem, a two‐dimensional Riemann problem and a bubble explosion under free surface, are used to examine its performance in solving the various compressible multi‐fluid flow problems with either the same types of EOS or different types of EOS. The results are compared with those calculated by the following methods: the method with ROE scheme (J. Comput. Phys. 2002; 118 :577–616), the seven equation model (J. Comput. Phys. 1999; 150 :425–467), Shyue's fluid‐mixture model (J. Comput. Phys. 2001; 171 :678–707) or the method in Liu et al. (Comp. Fluids 2001; 30 :315–337). The comparisons for the test problems show that the proposed method seems to be more accurate than the method in Allaire et al. (J. Comput. Phys. 2002; 118 :577–616) or the seven‐equation model (J. Comput. Phys. 1999; 150 :425–467). They also show that it can adaptively and accurately solve these compressible multi‐fluid problems and preserve the oscillation‐free property of pressure and velocity across the material interface. Copyright © 2010 John Wiley & Sons, Ltd.     

Comments on ‘Simplex space–time meshes in finite element simulations’

Some comments are provided on the citations offered in a recent paper (M. Behr, Int. J. Numer. Meth. Fluids 2008; 57 :1421–1434) that describes space–time finite element computations of advection of ‘Gaussian hills’, including computations with mesh refinement in the time direction. Copyright © 2008 John Wiley & Sons, Ltd.     

QALE‐FEM for modelling 3D overturning waves

A further development of the QALE‐FEM (quasi‐arbitrary Lagrangian–Eulerian finite element method) based on a fully nonlinear potential theory is presented in this paper. This development enables the QALE‐FEM to deal with three‐dimensional (3D) overturning waves over complex seabeds, which have not been considered since the method was devised by the authors of this paper in their previous works (J. Comput. Phys. 2006; 212 :52–72; J. Numer. Meth. Engng 2009; 78 :713–756). In order to tackle challenges associated with 3D overturning waves, two new numerical techniques are suggested. They are the techniques for moving the mesh and for calculating the fluid velocity near overturning jets, respectively. The developed method is validated by comparing its numerical results with experimental data and results from other numerical methods available in the literature. Good agreement is achieved. The computational efficiency of this method is also investigated for this kind of wave, which shows that the QALE‐FEM can be many times faster than other methods based on the same theory. Furthermore, 3D overturning waves propagating over a non‐symmetrical seabed or multiple reefs are simulated using the method. Some of these results have not been found elsewhere to the best of our knowledge. Copyright © 2009 John Wiley & Sons, Ltd.     

Probabilistic Analysis of the Upwind Scheme for Transport Equations

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the scheme is of order 1/2 in L^￥([0,T],L¹(\mathbb R^d)){L^{\infty}([0,T],L^1(\mathbb R^d))} for an integrable initial datum of bounded variation and of order 1/2−ε, for all ε > 0, in L^￥([0,T] ×\mathbb R^d){L^{\infty}([0,T] \times \mathbb R^d)} for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.     

An evaluation of a 3D free‐energy‐based lattice Boltzmann model for multiphase flows with large density ratio

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free‐energy‐based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56 :1653–1671) is applied to solve the 3D C–H equation. To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long‐time simulation of a bubble rising. How to handle the inlet/outlet and moving‐wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation. This model's performance on two‐phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high‐Re cases, the mass conservation seems not so good as the low‐Re case. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical investigation of the barotropic instability on the equatorial β-plane

The barotropic instability of horizontal shear flows is investigated by using two numerical algorithms to solve the equatorial β-plane barotropic equations. The first is the Arakawa Jacobian method (Arakawa, in J Comput Phys 1:119–143, 1966), which is a second-order-centered finite differences scheme that conserves energy and enstrophy, and the second is the fourth-order essentially non-oscillatory scheme for non-linear PDE’s of Osher and Shu (SIAM J Numer Anal 28:907–922, 1991), which is designed to track sharp fronts. We are interested in the performance of these two methods in tracking the long-time behavior of the instability, under the influence of the non-linearity, in the simple case of a Helmholtz shear layer. The associated linear problem is solved analytically, and the linear solution is used as an initial condition for the numerical simulations. We run a series of numerical simulations using both methods with various grid refinements and with two different amplitudes of the initial perturbation. A small viscosity term is added to the vorticity equation to damp the grid-scale waves for Arakawa’s method. This is not necessary for the high-order ENO-4 scheme, which has its own grid-scale dissipation. At high resolution, the two methods are in good agreement; they yield qualitatively and quantitatively the same solution in the long run: for small disturbances, the total flow stabilizes into a steady-state meridional shear with a smooth profile near the equator, while strong disturbances merge together to form a single large-scale vortex that propagates westward, along the equator, consistent with the African easterly waves and the monsoons trough circulation. At coarse resolution, however, Arakawa’s method seems to be much superior to the fourth-order ENO-4 scheme as it provides solutions that are more consistent with the fine resolution one.     

The general boundary element method and its further generalizations

In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publications, Southampton, MA, 1995, pp. 67–74; Int. J. Numer. Methods Fluids, 23 , 739–751 (1996), 24 , 863–873 (1997); Comput. Mech., 20 , 397–406 (1997)] and Liao and Chwang [Int. J. Numer. Methods Fluids, 23 , 467–483 (1996)] are further generalized by introducing a non‐zero parameter . Some related mathematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non‐linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermore, the general BEM can solve non‐linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative methodology of the BEM for non‐linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non‐linear problems are used to illustrate the validity and potential of the further generalized BEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Influence of mesh deformation on the quality of large eddy simulations

The influence of mesh motion on the quality of large eddy simulation (LES) was studied in the present article. A three‐dimensional, turbulent pipe flow (Re_τ=360) was considered as a test case. Simulations with both stretching and static meshes were carried out in order to understand how mesh motion affects the turbulence statistics. The spatial filtering of static and moving mesh direct numerical simulation (DNS) data showed how an ideal LES would perform, while the comparison of DNS cases with static and moving meshes revealed that no significant numerical errors arise from the mesh motion when the simulation is fully resolved. The comparison of the filtered fields of the DNS with a moving mesh with the corresponding LES fields revealed different responses to mesh motion from different numerical approaches. A straightforward test was applied in order to verify that the moving mesh works consistently in LES: when the mesh is stretched in the streamwise direction, the moving mesh results should be in between the two extremal resolutions between which the mesh is stretched. Numerical investigations using four different LES approaches were carried out. In addition to the Smagorinsky model, three implicit LES approaches were used: linear interpolation (non‐dissipative), the Gamma limiter (dissipative), and the scale‐selective discretisation (slightly dissipative). The results indicate that while the Smagorinsky and the scale‐selective discretisation approaches produce results consistent with the resolution of the non‐static mesh, the implicit LES with linear interpolation or the Gamma scheme do not. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive mesh refinement techniques for high‐order shock capturing schemes for multi‐dimensional hydrodynamic simulations

The numerical simulation of physical phenomena represented by non‐linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a method obtained by the combination of a high‐order shock capturing scheme, built from Shu–Osher's conservative formulation (J. Comput. Phys. 1988; 77 :439–471; 1989; 83 :32–78), a fifth‐order weighted essentially non‐oscillatory (WENO) interpolatory technique (J. Comput. Phys. 1996; 126 :202–228) and Donat–Marquina's flux‐splitting method (J. Comput. Phys. 1996; 125 :42–58), with the adaptive mesh refinement (AMR) technique of Berger and collaborators (Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. Thesis, Computer Science Department, Stanford University, 1982; J. Comput. Phys. 1989; 82 :64–84; 1984; 53 :484–512). Copyright © 2006 John Wiley & Sons, Ltd.     

Some Indentities for the Gradient of the Principal Invariants, Traces and Determinants via Grassmann Calculus

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants of A and obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.     

Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd.     

Global Solvability of a Free Boundary Three-Dimensional Incompressible Viscoelastic Fluid System with Surface Tension

Motivated by Beale (Commun Pure Appl Math 34:359–392, 1981; Arch Ration Mech Anal 84:307–352, 1983/1984), we investigate the global well-posedness of a free boundary problem of a three-dimensional incompressible viscoelastic fluid system in an infinite strip and with surface tension on the upper free boundary, provided that the initial data is sufficiently close to the equilibrium state.     

Lyapunov and Sacker–Sell Spectral Intervals

In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some restrictions– the Lyapunov spectral intervals. Since any bounded and continuous coefficient matrix function can be smoothly transformed to an upper triangular matrix function, our results imply that these spectral intervals may be found from scalar homogeneous problems. In line with our previous work [Dieci and Van Vleck (2003), SIAM J. Numer. Anal. 40, 516–542], we emphasize the role of integral separation. Relationships between different spectra are shown, and examples are used to illustrate the results and define types of coefficient matrix functions that lead to continuous Sacker–Sell spectrum and/or continuous Lyapunov spectrum.      

Orthogonality of modal bases in hp finite element models

In this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20 :1671–1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least‐squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L₂ least‐squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward‐facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein. Copyright © 2007 John Wiley & Sons, Ltd.     

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.