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This paper presents a finite element method for the simulation of compressible flows. The Navier–Stokes and Euler equations are solved in the conservation form using various sets of independent variables. A variational formulation is developed based upon a variant of the Petrov–Galerkin method, and uses a shock-capturing operator. An adaptive algorithm based on a particular residual norm is proposed. Several numerical examples are presented to demonstrate the performances of each set of variables in solving compressible high-speed flows.

When attempting to compute unsteady, variable density flows at very small or zero Mach number using a standard finite volume compressible flow solver one faces at least the following difficulties: (i) Spatial pressure variations vanish as the Mach number M->0, but they do affect the velocity field at leading order; (ii) the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux; (iii) violations of this constraint crucially affect the transport of mass, preventing a code to properly advect even a constant density distribution. We overcome these difficulties through a new algorithm for constructing numerical fluxes in the context of multi-dimensional finite volume methods in conservation form. The construction of numerical fluxes involves: (1) An explicit upwind step yielding predictions for the nonlinear convective flux components. (2) A first correction step that introduces pressure gradients which guarantee compliance of the convective fluxes with a divergence constraint. This step requires the solution of a first Poisson-type equation. (3) A second projection step which provides the yet unknown (non-convective) pressure contribution to the total flux of momentum. This second projection requires the solution of another Poisson-type equation and yields the cell centered velocity field at the new time. This velocity field exactly satisfies a divergence constraint consistent with the asymptotic limit. Step (1) can be done by any standard finite volume compressible flow solver. The input to steps (2) and (3) involves solely the fluxes from step (1) and is independent of how these were obtained. Thus, our approach allows any such solver to be extended to compute variable density incompressible flows.

A numerical method of solution is developed for steady–state hyperbolic equations in conservation form. An upwind direction is identified and a smooth mapping is introduced to align the equations with that direction. A flux F ( u ) in the upwind direction is regarded as the ‘state’ variable and advanced numerically on a grid according to a second–order extension of Godunov' method. The solution u is determined from the computed flux, and a constraint on the mapping is that the Jacobian, det( ∂F ∂u ), is of one sign. The method is applied to the equations modelling shock–wave propagation according to Whitham' theory of geometrical shock dynamics. In this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times. Shock propagation in both uniform and non–uniform gases is considered and several example calculations are presented. A further application of the numerical method is inviscid supersonic flow, as governed by the steady Euler equations. The steady supersonic flow in a converging–diverging duct is computed to illustrate the numerical method for this latter application.

A flow model is presented for predicting a hydraulic jump in a straight open channel. The model is based on the general 2D shallow water equations in strong conservation form, without artificial viscosity, which is usually incorporated into the flow equations to capture a hydraulic jump. The equations are discretised using the finite volume method. The results are compared with experimental data and available numerical results, and have shown that the present model can provide good results. The model is simple and easy to implement. To demonstrate the potential application of the model, several hydraulic jumps occurring in different situations are simulated

An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the AMRCLAW package, which is freely available.

An iterative implicit-explicit hybrid scheme is proposed for one-dimensional ideal magnetohydrodynamical (MHD) equations. The scheme is accurate to second order in both space and time for all Courant numbers. Each kind of MHD wave, both compressible and incompressible, may be either implicitly or explicitly, or partially implicitly and partially explicitly treated depending on their associated Courant numbers in each numerical cell, and the scheme is able to smoothly switch between explicit and implicit calculations. The scheme is of Godunov type in both explicit and implicit regimes, in which the flux needed in a Godunov scheme is calculated from Riemann problems, is in a strictly conservation form, and is able to resolve MHD discontinuities. Only a single level of iterations is involved in the scheme, and the iterations solve both the implicit relations arising from upstream centered differences for all wave families and the nonlinearity of MHD equations. Multicolors proposed in the paper may significantly reduce the number of iterations required to reach a converged solution. Compared with the largest Courant number in a simulation, only a small number of iterations are needed in the scheme. The feature of the scheme has been shown through numerical examples. It is easy to vectorize the computer code of the scheme.

This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes defined on non-Cartesian grids. We identify those schemes which are consistent and prove that the $L^\infty(0,T;L^1(\Bbb{R}^d))$-norm of the error goes to zero as $(\Delta x)^{1/2}$ when the discretization parameter $\Delta x$ goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of $(\Delta x)^{1/2}$ because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.

The effect of using grid adaptation on the numerical solution of model convection-diffusion equations with a conservation form is studied. The grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc-length of the solution. In particular, a parameter-dependent monitor function is introduced which incorporates fixed meshes, approximate arc-length equidistribution, and equidistribution of the absolute value of the solution, in a single framework. Thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values. A class of singularly perturbed problems, including Burgers's equation in the limit of small viscosity, is studied. Singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations, and numerical results are compared with the results from the analysis. Computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory. It is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that, furthermore, the spurious solutions can be stable for an appropriate moving mesh method.

SUMMARY In this paper, alternative formulations of the steady Euler equations for conservation of mass, momentum and energy are adopted for the numerical simulation of compressible flows with shock waves. The total enthalpy is assumed to be constant and hence an isentropic density is calculated in terms of the velocity components. Also, the x- and y-momentum equations written in conservation form are combined to yield the tangential and normal momentum equations. For smooth flows the tangential momentum equation reduces to the entropy transport equation, while the normal momentum equation gives the vorticity in terms of the entropy gradient normal to the flow direction (Crocco’s relation). Hence the velocity components can be obtained from the continuity equation and normal momentum equation (Cauchy=Riemann equations), while the entropy correction for the density is obtained from the tangential momentum equation (this correction is not needed in the isentropic flow regions). The present formulation can be easily extended to handle variable total enthalpy. Preliminary results are presented for transonic and supersonic flows over aerofoils and the entropy and vorticity effects are clearly identified. # 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids, 27: 127‐138 (1998)

The features of an axial-symmetric two-dimensional (2D) model in a transient cavitating pipe flow are investigated. A distributed vaporous cavitation model is proposed, based on the mass and momentum balance equations for a liquid-vapor mixture. A conservation form of the continuity equation allows a simple numerical solution. Both one-dimensional (1D) and 2D models are considered to quantify the effect of friction in the simulation of experimental data. The axial-symmetric 2D model allows the evaluation of the velocity profile and a more accurate estimate of the wall shear stress. The comparison between the results of numerical runs and experimental data of pressure head oscillations in transient cavitating pipe flows shows that the errors on maximum head oscillations of 2D model are generally greatly reduced with respect to those of the 1D model. As expected, the quasi-steady 1D model does not adequately represent the experimental data, with exception of the first maximum oscillations, whereas t...

A high-order Godunov-type scheme is developed for the shock interactions in ideal magnetohydrodynamics (MHD). The scheme is based on a nonlinear Riemann solver and follows the basic procedure in the piecewise parabolic method. The scheme takes into account all the discontinuities in ideal MHD and is in a strict conservation form. The scheme is applied to numerical examples, which include shock-tube problems in ideal MHD and various interactions between strong MHD shocks. All the waves involved in the corresponding Riemann problems are resolved and are correctly displayed in the simulation results. The correctness of the scheme is shown by the comparison between the simulation results and the solutions of the Riemann problems. The robustness of the scheme is demonstrated through the numerical examples. It is shown that the scheme offers the principle advantages of a high-order Godunov-type scheme: robust operation in the presence of very strong waves, thin shock fronts, and thin contact and slip surface discontinuities.

This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme. The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall. The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.

We present in two parts, a mathematical theory of conservation laws using the language of physics. In Part I we explain the concept of a special type of nonlinearity which appears in an important class of evolutionary processes governed by hyperbolic partial differential equations. For simplicity, we develop the theory using a simple model equation. We show that it is possible to extend the concept of solutions with discontinuities with the help of a conservation form of the equation.

A class of high resolution multidimensional wave-propagation algorithms is described for general time-dependent hyperbolic systems. The methods are based on solving Riemann problems and applying limiter functions to the resulting waves, which are then propagated in a multidimensional manner. For nonlinear systems of conservation laws the methods are conservative and yield good shock resolution. The methods are generalized to hyperbolic systems that are not in conservation form and to problems that include a “capacity function.” Several examples are included for gas dynamics, acoustics in a heterogeneous medium, and advection in a stratified flow on curvilinear grids. The software packageCLAWPACKimplements these algorithms in Fortran and is freely available on the Web. One and two space dimensions are discussed here, although the algorithms and software have also been extended to three dimensions.

This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of ( Δ x ) 1 / 2 (\Delta x)^{1/2} in L ∞ ( L 1 ) L^{\infty }(L^{1}) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p − p- consistent schemes, and prove that they converge to the entropy solution with the rate of ( Δ x ) min { 1 / 2 , p } (\Delta x)^{\min \{1/2,p\}} in L ∞ ( L 1 ) L^{\infty }(L^{1}) ; again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of ( Δ x ) 1 / 2 (\Delta x)^{1/2} in L ∞ ( L 1 ) L^{\infty }(L^{1}) . We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit Rankine–Hugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( $$\sqrt t $$ ) for large time, and the front fluctuation scaled by $$\sqrt t $$ converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root t normalized front standard deviation in terms of the random media statistics. The theory agrees remarkably with the numerically discovered empirical formula.

A shock-capturing, finite volume implementation of recently proposed non-hydrostatic two-dimensional shallow water equations, is proposed. The discretization of the equations in conservation form implies the modification of the time derivative of the conserved variable, in the form of a mass/inertia matrix, and extra terms in the flux functions. The effect of this matrix is to slow down wave propagation in the presence of significant bottom slopes. The proposed model is first derived in conservation form using mass and momentum balance principles. Its finite volume implementation is then presented. The additional terms to the shallow water equations can be discretized very easily via a simple time-stepping procedure. Two application examples are presented. These examples seem to indicate that the proposed model does not exhibit strong differences with the classical hydrostatic shallow water model under steady-state conditions, but that its behavior is significantly different when transients are involved.

An accurate and robust parallel implicit adaptive mesh re nement (AMR) algorithm is proposed and described for the prediction of unsteady behaviour of laminar ames. The scheme is applied to the solution of the system of the partial-di erential equations governing time-dependent, three-dimensional, compressible laminar ows for reactive thermally perfect gaseous mixtures. A high-resolution nite-volume spatial discretization procedure is used to solve the conservation form of these equations on bodytted multi-block hexahedral mesh. A local preconditioning technique is used to remove numerical sti ness and maintain solution accuracy for low-Mach-number, nearly incompressible ows. A exible block-based octree data structure has been developed and is used to facilitate automatic solution-directed mesh adaptation according to physics-based re nement criteria. The data structure also enables an e cient and scalable parallel implementation via domain decomposition. The parallel implicit formulation makes use of a dual-time-stepping like approach with an implicit second-order backward discretization of the physical time, in which a Jacobian-free inexact Newton method with a preconditioned generalized minimal residual (GMRES) algorithm is used to solve the system of nonlinear algebraic equations arising from the temporal and spatial discretization procedures. An additive Schwarz global preconditioner is used in conjunction with block incomplete LU type local preconditioners for each sub-domain. The Schwarz preconditioning and block-based data structure readily allow e cient and scalable parallel implementations of the implicit AMR approach on distributed-memory multi-processor architectures. Numerical results for steady and unsteady laminar coow di usion and premixed methane-air ames demonstrate the capabilities of the proposed approach for a range of reactiveow applications. The scheme is shown to accurately predict key characteristics of the di usion ames. For a premixed ame under terrestrially gravity, the scheme is also shown to accurately predict the frequency of the natural buoyancy induced oscillations.

Continuum kinetic models, such as Maxwell-Boltzmann, present a viable alternative to particle-in-cell (PIC) models because they can be cast in conservation form and are not susceptible to noise. By treating the associated phase space distribution function as a continuous incompressible fluid occupying a volume of position-velocity space, evolution of the distribution function is determined by solving a 6-D advection equation. In cases where collision terms are negligible, the Boltzmann model is reduced to a Vlasov model. A 4th-order accurate continuum kinetic Vlasov model has been developed in one spatial and one velocity dimension to address the challenges associated with solving a hyperbolic governing equation in multidimensional phase space. The governing equation is cast in conservation law form and solved with a finite volume representation. Adaptive mesh refinement (AMR) is used to allow for efficient use of computational resources while maintaining desired levels of resolution. Consequently, with AMR the model is able to capture the fine structures that develop in the distribution function as it evolves in time, while using low resolution in the tail of the distribution function. The model is tested on several plasma instability problems including: the two-stream instability and the beam-plasma instability. The model demonstrates conservation of mass in that the total integral of the distribution function is preserved, as well as the conservation of energy. Model extension into two and three spatial dimensions is discussed.

This paper describes a numerical simulation of the flow at the delta of the Po River, the main Italian river. We take into account a portion of the river that is 20 km long, characterized by a complex geometry consisting of narrow bends and strong gradients in the profile of the riverbed. To account for the presence of tidal effects, a nonstationary solution is sought. The considered model consists in the classical two-dimensional (2D) Saint-Venant equations, written in conservation form. This system of partial differential equations is discretized by a finite-element scheme that has some particularly attractive features for river flow. The accuracy of the scheme is verified on a few test cases, then a numerical simulation of the flow of the Po over three days is implemented. The numerical results are then compared with the experimental measures and discussed in the light of the assumptions made in the construction of the model.

The equations of motion for two-dimensional deformations of an incompressible elastoplastic material involve five equations, two equations expressing conservation of momentum, and three constitutive laws, which we take in the rate form, i.e. relating the stress rate to the strain rate. In hypoplasticity, the constitutive laws are homogeneous of degree one in the stress and strain rates. This property has the consequence that although the equations are not in conservation form, there is nonetheless a natural way to characterize planar shock waves. The Riemann problem is the initial value problem for plane waves, in which the initial data for stress and velocity consist of two constant vectors separated by a single discontinuity. The main result is that, under appropriate assumptions, the Riemann problem has a scale invariant piecewise constant solution. The issue of uniqueness is left unresolved. Indeed, we give an example satisfying the conditions for existence, for which there are many solutions. Using asymptotics, we show how solutions of the Riemann problem are approximated by smooth solutions of a system regularized by the addition of viscous terms that preserve the property of scale invariance.

We present a Chebyshev multidomain method that can solve systems of hyperbolic equations in conservation form on an unrestricted quadrilateral subdivision of a domain. Within each subdomain the solutions and fluxes are approximated by a staggered-grid Chebyshev method. Thus, the method is unstructured in terms of the subdomain decomposition, but strongly structured within the subdomains. Communication between subdomains is done by a mortar method in such a way that the method is globally conservative. The method is applied to both linear and nonlinear test problems and spectral accuracy is demonstrated.

An iterative implicit?explicit hybrid scheme is proposed for hyperbolic systems of conservation laws. Each wave in a system may be implicitly, or explicitly, or partially implicitly and partially explicitly treated depending on its associated Courant number in each numerical cell, and the scheme is able to smoothly switch between implicit and explicit calculations. The scheme is of Godunov-type in both explicit and implicit regimes, is in a strict conservation form, and is accurate to second-order in both space and time for all Courant numbers. The computer code for the scheme is easy to vectorize. Multicolors proposed in this paper may reduce the number of iterations required to reach a converged solution by several orders for a large time step. The feature of the scheme is shown through numerical examples.

The simulation of the scattering of plane electromagnetic waves by obstacles constructed from piecewise homogeneous material media is considered. The governing equations are Maxwell's equations, which are expressed in a conservation form. The solution is obtained on a grid consisting of an unstructured assembly of linear triangular elements in 2D, by modifying a technique developed originally for the solution of the compressible Euler equations. The computed solution is used to determine the radar scattering width of the obstacle.

Shallow-water flows with supercritical and subcritical subregions often exhibit numerical difficulties because of their associated hydraulic jumps (shock waves), steep layers, and fictitious oscillations. Analogous problems in gas dynamics have led to the recent development of a promising class of Petrov-Galerkin methods specifically designed for hyperbolic/incompletely parabolic systems, and are written in a symmetric conservation form. One of the major difficulties in the application of this class of methods to shallow water problems has been the unavailability of a suitable symmetric form of the governing equations. In the present work, this issue is addressed by introducing the total energy of the water column to motivate a change of variables which symmetrizes the shallow-water conservation system. Then, the one-dimensional case is considered and a time-accurate, streamline-upwind Petrov-Galerkin (SUPG) scheme is developed based on the proposed symmetric form. Numerical results illustrate the method and permit comparison with other schemes.

The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier--Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier--Stokes equations and utilizes linearization and localization at the boundaries based on these variables. The proposed boundary conditions are applied through a penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity. The versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder.

In this paper we study various aspects of gravity (or density) currents arising from instantaneous releases of heavy fluids in a rectangular channel with a horizontal bottom. It is shown, by means of a scaling argument, that these plane currents can be successfully modeled by a two-by-two system in conservation form together with a pair of algebraic relations. A number of numerical experiments are carried out using this “weak stratification” model to elicit information concerning the behavior of gravity currents. A weakly nonlinear analysis is employed to clarify some aspects that were uncovered by the numerical experimentation.

We replace the flux term in Burger's equation by two simple alternates that contain contributions depending globally on the solution. In one case, the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal, and in the other the term is in conservation form. In both cases, the nonanalytic is due to the presence of the Hilbert transform. The equations have a loose analogy to the motion of vortex sheets. In particular, they both form singularities in finite time in the absence of viscous effects. Our motivation then is to study the influence of viscosity. In one case, viscosity does not prevent singularity formation. In the other, we can prove solutions exist for all time, and determine the likely weak solution as viscosity vanishes. An interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time. In one case, the singularity is a pole and causes the solution to blow up when it reaches the real axis. In the other, numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time, and then propagates along it. We hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations.

Multilevel, finite element discretization methods for the Navier–Stokes equations are considered. In contrast to usual multilevel methods, a superlinear scaling of the consecutive meshwidths $h_J + 1 = \mathcal {O}(h_j^{\alpha (j)} )$ is used. On the coarsest mesh the discretized system is solved, which is small and nonlinear. On each subsequent mesh only one or two Newton correction steps are performed, so that only one or two larger, linear systems are solved. The scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the $H^1 $- and $L^2 $-norm are investigated.An error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term. This formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew-symmetric corms are used for the nonlinear term.

This paper generalizes the Ericksen-Leslie continuum model of liquid crystals to allow for dynamically evolving line defect distributions. In analogy with recent mesoscale models of dislocations, we introduce fields that represent defects in orientational and positional order through the incompatibility of the director and deformation ‘gradient’ fields. These fields have several practical implications: first, they enable a clear separation between energetics and kinetics; second, they bypass the need to explicitly track defect motion; third, they allow easy prescription of complex defect kinetics in contrast to usual regularization approaches; and finally, the conservation form of the dynamics of the defect fields has advantages for numerical schemes. We present a dynamics of the defect fields, motivating the choice physically and geometrically. This dynamics is shown to satisfy the constraints, in this case quite restrictive, imposed by material-frame indifference. The phenomenon of permeation appears as a natural consequence of our kinematic approach. We outline the specialization of the theory to specific material classes such as nematics, cholesterics, smectics and liquid crystal elastomers. We use our approach to derive new, non-singular, finite-energy planar solutions for a family of axial wedge disclinations.

A recently developed numerical model for environmental flows has been applied to simulate the water circulation and pollutant dispersion in a confined coastal system in Cyprus. The model is based on the numerical solution of the time averaged Navier-Stokes equations, written in their contravariant strong conservation form, in a generalised non-orthogonal coordinate system. The model predictions are verified compared with experimental data incorporating radioactive tracer injection at the surface of the sea, then the model is applied to predict the dispersion of waste from the outfall pipe of a sewage treatment plant. The predicted results agree favourably with the measurements, demonstrating the code's validity, flexibility and economy for predicting accurately flows in three-dimensional complex sea bed terrain basins. The utilisation of such a model can be an advantageous alternative to the expensive and time consuming experimental field work, enabling the study of the plume dispersion under different discharge and hydrological prevailing conditions.

The two-dimensional shallow-water flow equations are first written in strong conservation form and are used for developing the flow model. In the discretization equations, based on the control volume method, the velocity field is found to be proportional to both the square of the depth and the depth itself. Due to this, none of the existing SIMPLE-like algorithms can be used to deal with the velocity-depth coupling. In this paper, the problem of velocity-depth coupling is studied, and a modified SIMPLE-like algorithm is described to treat it. Several applications of the model have shown that the method is efficient, simple, and has good convergence behavior. Also, a different value for the relaxation factor for depth is found to speed the convergence rate after stable convergence occurs. The results are in good agreement with both experimental observations and theoretical analysis. The method can straightforwardly be extended into unsteady shallow-water flows.

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge-Kutta time discretizations. A constrained transport-type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.

In this work we present a new approach to the construction of high order finite volume central schemes on staggered grids for general hyperbolic systems, including those not admitting a conservation form. The method is based on finite volume space discretization on staggered cells, central Runge--Kutta time discretization, and integration over a family of paths, associated to the system itself, for the generalization of the method to nonconservative systems. Applications to the one- and two-layer shallow water models as prototypes of systems of balance laws and systems with source terms and nonconservative products, respectively, will be illustrated.

We shall call the system of coordinates (λ, ξ, η, ς) defined in (6.1) unified in the sense that it unifies the Eulerian system when Q = 0 with the Lagrangian when Q = q, and also in the sense that the system of governing equations (6.19) unites the geometrical conservation laws with the physical ones to form a closed system of PDE in conservation form.

Fully nonlinear modulation equations are presented that govern the slow evolution of single-phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows, among which is the discrete nonlinear Schrodinger equation (DNLS). These modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data. In some cases these equations are capable of dynamically changing type from hyperbolic to elliptic, a phenomenon that has been associated with the modulational instability of the underlying wavetrain. By putting the modulation equations in Riemann invariant form, one can select initial data that avoid this dynamic change of type. Numerical experiments demonstrating the theoretical results are presented.

The solution of problems involving the scattering of plane electromagnetic waves by perfectly conducting obstacles is considered. The governing equations are Maxwell's equations, which are expressed in a conservation form. The solution is then obtained by modifying a technique which was applied originally to the solution of the compressible Euler equations on a mesh consisting of an unstructured assembly of linear triangular elements in 3D. The computed solution can be used to determine the radar cross-section of the scatterer.

A finite element method for solving the compressible Navier-Stokes equations is presented. These equations are solved in conservation form and using conservative variables. Appropriate finite element approximations are discussed when a Galerkin variational formulation is used. For high-speed flows, the formulation is stabilized using the stream line upwind Petrov-Galerkin method for which we propose an analytical expression for the matrix τ. A new discontinuity-capturing operator is also proposed to accurately solve flow problems exhibiting sharp gradients. The non-linear systems of equations arising from the discretization are solved using an iterative strategy based on the generalized minimal residual (GMRES) algorithm. Numerical results for transonic and supersonic flows are presented that demonstrate the effectiveness of the proposed finite element method.

A numerical method is called robust if it does not require tuning. Two different approaches to robustness are described. First, existing energy and enstrophy conserving space discretizations are combined with new nonlinear Runge-Kutta schemes that conserve either energy or enstrophy in the inviscid unforced case. This guarantees unconditional stability of long-term integrations, even for the explicit variants. The explicit nonlinear Runge-Kutta schemes are inaccurate (but stable) if the related standard Runge-Kutta formula is unstable, Fortunately, the nonlinear schemes yield a parameter which can be used to signal inaccuracies and to adapt the timestep such that results are accurate. However, as they are bead on symmetric space discretizations, these methods still require the ad hoc tuning of an artificial viscosity to suppress numerical spatial oscillations. The second approach is to construct robust space discretizations (Jacobians) using the grid-aligned essentially non-oscillatory (ENO) technique. ENO Jacobians based on the conservation form of the vorticity equation generate spikes near stagnation points. Only fifth and lower order ENO Jacobians based on the advection form of the vorticity equation suppress all instabilities and oscillations without tuning if a global Lax-Friedrichs modification is used. A preliminary comparison is made with a tuned Arakawa Jacobian.

A recently developed mesoscale numerical model has been applied to the Athens Basin in order to provide a detailed description of the mean flow and pollution levels under neutral atmospheric conditions. The model is based on the numerical solution of the time-averaged Navier-Stokes equations written in their contravariant strong conservation form in a generalized non-orthogonal co-ordinate system. Before its application to the Athens Basin case, the reliability of the model was tested by predicting the flow and concentration fields over a three-dimensional hill. The predicted results agree favourably with available experimental data, demonstrating the validity, flexibility and economy of the model for flows in three-dimensional complex terrain

In this paper, the new Lagrangian method introduced by Loh and Hui is extended for 3D steady supersonic flow computation. We present the derivation of the conservation form and the solution of the local Riemann solver using the Godunov and the high resolution TVD schemes. The new approach is accurate and robust, capable of handling complicated geometry and interactions between discontinuous waves. As shown in the test problems, the current Lagrangian method retains all the advantages claimed in the 2D method, e.g., crisp resolution of a slip-surface (contact discontinuity) and automatic grid generation. In this paper, we also suggest a novel 3D Riemann problem in which interesting and intricate flow features are present.

In this paper, a class of essentially conservative scheme are constructed and analyzed. The numerical tests and theoretical analysis show that although these schemes can not be written in the usual conservation form but the numerical solutions obtained with these schemes can converge, as the mesh size tends to zero, to the physical solution of conservation laws.

The selection of primary dependent variables for the solution of Navier-Stokes equations in the curvilinear body-fitted coordinates is still an unsettled issue. Reported formulations with primitive variables involve contravariant velocity components, Cartesian components, and velocity projections, also known as resolutes. Most of the formulations result in a weak conservation form of the momentum equations which contain grid line curvature- and divergence-related Coriolis and centrifugal terms. This paper presents a general strong conservation formulation of the momentum equations allowing the flexibility in choosing the various forms of the velocity components as the dependent variables. Ambiguous issues relating geometrical topology and forms of governing equations are discussed and clarified. Computational results obtained with both strong and weak forms are presented and compared to known analytical/experimental data. The results confirm the soundness of the formulation. 19 refs.

The dynamics of inviscid multicomponent fluids may be modelled by the Euler equations, augmented by one (or more) additional species equation(s). Attempts to compute solutions for extended Euler models in conservation form, show strong oscillations and other computational inaccuracies near material interfaces. These are due to erroneous pressure fluctuations generated by the conservative wave model. This problem does not occur in single component computations and arises only in the presence of several species. A nonconservative (primitive) Euler formulation is proposed, which results in complete elimination of the oscillations. The numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation. Numerical experiments show clean monotonic solution profiles, with acceptably small conservation error for shocks of weak to moderate strengths.

The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections. These quantum corrections are $O( {\hbar ^2 } )$. The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation. The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms. These quantum terms allow particles to tunnel through potential barriers and to build up in potential wells.The three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode. The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind method.Simulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v...

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.

An analysis is made on diffraction of plane waves by a corner. Basic equations of the analysis are linearized gasdynamic equations which are considered to be valid for flow fields induced by weak shock waves. Analytical expressions are obtained for velocity fields as well as density fields of diffraction of plane waves by a corner. Numerical examples of the analysis show that the velocity tends toward infinity at the apex of the corner although neither the density nor the pressure has singularity there. Numerical simulations are conducted for diffraction of weak shock waves on a corner at 270 degrees employing the explicit TVD finite difference method in conservation form. The analytical results are compared with computed results of simulations. Good agreement is obtained between them except in the neighborhood of the apex of the corner. In the computed flow fields, a vortex is observed to be generated and attached just behind the apex of the corner.

A Runge–Kutta Discontinuous Galerkin method (RKDG) to solve the reactive Navier–Stokes equations written in conservation form and with no restrictive physical hypothesis is presented. Real thermodynamics laws are taken into account. A particular care has been taken to solve correctly the stiff gaseous interfaces. Some 1-D and 2-D test cases are presented.

A numerical solution has been found for the hyperbolic initial-value problem, which follows from the 16-moment set of equations describing the transport of a plasma in a strong magnetic field. In addition to the time-dependent evolution of mass and momentum, these equations describe the variations in anisotropic temperatures and heat fluxes for both ion and electron species. The numerical solution employs a high-order Godunov method to achieve third-order accuracy in space and second-order accuracy in time. The method includes an approximate Riemann solver which is suitable for a system of equations that cannot entirely be expressed in conservation form. In addition, a new technique has been developed to overcome the stiffness that occurs in regions where plasma flows are strongly reactive. Simple test cases showing correct wave behavior are presented.