On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes.Because they do not involve any special data structure,and do not induce savings in memory requirements,they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required.The multilevel technique can also be applied to balance laws,but in this case,numerical errors may be induced by the technique.We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation,and leads to a more robust multilevel code for balance laws,while maintaining the efficiency rates observed forhyperbolic conservation laws.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.

Tensile Shock Physics in Compressible Thermoviscoelastic Solid Medium

This paper addresses tensile shock physics in thermoviscoelastic (TVE) solids without memory. The mathematical model is derived using conservation and balance laws (CBL) of classical continuum mechanics (CCM), incorporating the contravariant second Piola-Kirchhoff stress tensor, the covariant Green’s strain tensor, and its rates up to order n. This mathematical model permits the study of finite deformation and finite strain compressible deformation physics with an ordered rate dissipation mechanism. Constitutive theories are derived using conjugate pairs in entropy inequality and the representation theorem. The resulting mathematical model is both thermodynamically and mathematically consistent and has closure. The solution of the initial value problems (IVPs) describing evolutions is obtained using a variationally consistent space-time coupled finite element method, derived using space-time residual functional in which the local approximations are in hpk higher-order scalar product spaces. This permits accurate description problem physics over the discretization and also permits precise a posteriori computation of the space-time residual functional, an accurate measure of the accuracy of the computed solution. Model problem studies are presented to demonstrate tensile shock formation, propagation, reflection, and interaction. A unique feature of this research is that tensile shocks can only exist in solid matter, as their existence requires a medium to be elastic (presence of strain), which is only possible in a solid medium. In tensile shock physics, a decrease in the density of the medium caused by tensile waves leads to shock formation ahead of the wave. In contrast, in compressive shocks, an increase in density and the corresponding compressive waves result in the formation of compression shocks behind of the wave. Although these are two similar phenomena, they are inherently different in nature. To our knowledge, this work has not been reported in the published literature.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (X) -MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether＇s theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived.

Knowledge Law and Attribute Disturbance of Law

By employing the knowledge（R-element equivalence class） in one direction S-rough sets and dual of one direction S-rough sets, the concept of knowledge law is given; the generation theorem of knowledge law, the excursion theorem of knowledge law, and the attribute disturbance discernible theorem of knowledge law are proposed. Knowledge law is a new characteristic of S-rough sets.

ASYMPTOTIC RAREFACTION WAVES FOR BALANCE LAWS WITH STIFF SOURCES

We study the long time formation of rarefaction waves appearing in balance laws by means of singular perturbation methods. The balance laws are non standard because they contain a variable u that appears only in the flux terms. We present a concrete example occurring in flow of steam, nitrogen and water in porous media and an abstract example for a class of systems of three equations. In the concrete example the zero-order equations resulting from the expansion yield a type of conservation law system called compositional model in Petroleum Engineering. In this work we show how compositional models originate from physically more fundamental systems of balance laws. Under appropriate conditions, we prove that certain solutions of the system of balance laws decay with time to rarefaction wave solutions in the compositional model originating from the system of balance laws.

RIGOROUS ESTIMATES ON BALANCE LAWS IN BOUNDED DOMAINS

The initial-boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in [4], striving to provide a rigorous treatment and detailed references.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.

LARGE TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR VISCOELASTIC MODEL WITH FADING MEMORY

We study the Cauchy problem of a one-dimensional nonlinear viscoelastic model with fading memory. By introducing appropriate new variables we convert the integro-partial differential equations into a hyperbolic system of balance laws. When it is a perturbation of a constant state, the solution is shown time asymptotically approach- ing to predetermined diffusion waves, Pointwise estimates on the convergence details are obtained.

High-order accurate solutions of generalized Riemann problems of nonlinear hyperbolic balance laws

We provide a systematic study for the generalized Riemann problem(GRP)of the nonlinear hyperbolic balance law,which is critically concerned with the construction of the spatial-temporally coupled high-order Godunov-type scheme.The full analytical GRP solvers up to the third-order accuracy and also a collection of properties of the GRP solution are derived by resolving the elementary waves.The resolution of the rarefaction wave is a crucial point,which relies on the use of the generalized characteristic coordinate(GCC)to analyze the solution at the singularity.From the analysis on the GCC,we derive for the general nonlinear system the evolutionary equations for the derivatives of generalized Riemann invariants.For the nonsonic case,the full set of spatial and temporal derivatives of the GRP solution at the singularity are obtained,whereas for the sonic case the limiting directional derivatives inside the rarefaction wave are derived.In addition,the acoustic approximation of the analytical GRP solver is deduced by estimating the error it introduces.It is shown that the computationally more efficient Toro-Titarev solver can be the approximation of the analytical solver under the suitable condition.Hence this work also provides a theoretical basis of the approximate GRP solver.The theoretical results are illustrated via the examples of the Burgers equation,the shallow water equations and a system for compressible flows under gravity acceleration.Numerical results demonstrate the accuracy of the GRP solvers for both weak and strong discontinuity cases.

On the Special Application of the Principle of Proportionality under Emergency State

As the two fundamental aspects of modern society,the emergency state and the routine state are not exceptions to the rule of law.They just abide by special legal rules and must adhere to the spirit of the rule of law,such as human rights protection and power restrictions and apply the principle of proportionality.In a state of emergency,public interests are faced with major and urgent threats.In this state,the positioning of the purpose,the examination of the consequences,or the measurement of the legal benefits of the purpose and the consequences all have a certain degree of particularity and complexity.In order to increase the rationality of the principle of proportionality in the state of emergency,and to perform its value function scientifically and effectively,it is necessary to adopt loose application standards based on the characteristics of the state of emergency,grasp the key application links,and limit the scope of application to the review of the rationality of the exercise of emergency powers.Judgment of the legitimacy of the purpose of the exercise of emergency powers and the derogation of civil rights such as human dignity are not within the scope of the principle of proportionality.

A New Approach of High OrderWell-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element methods.We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is provided.The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.The same idea can be applied to the finite volume WENO schemes.We will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.

Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis:Global Dynamics and Diffusivity Limit

In this paper,we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate.Utilizing energy methods,we show that under time-dependent Dirichlet boundary conditions,long-time dynamics of solutions are driven by their boundary data,and there is no restriction on the magnitude of initial energy.Moreover,the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions,which has not been observed in previous studies on related models.

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

With increasing engineering demands,there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct“physics”.There are two families of high order methods:One is the method of line,relying on the Runge-Kutta(R-K)time-stepping.The building block is the Riemann solution labeled as the solution element“1”.Each step in R-K just has first order accuracy.In order to derive a fourth order accuracy scheme in time,one needs four stages labeled as“1111=4”.The other is the one-stage Lax-Wendroff(LW)type method,which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems.In recent years,the pair of solution element and dynamics element,labeled as“2”,are taken as the building block.The direct adoption of the dynamics implies the inherent temporal-spatial coupling.With this type of building blocks,a family of two-stage fourth order accurate schemes,labeled as“22=4”,are designed for the computation of compressible fluid flows.The resulting schemes are compact,robust and efficient.This paper contributes to elucidate how and why high order accurate schemes should be so designed.To some extent,the“22=4”algorithm extracts the advantages of the method of line and one-stage LW method.As a core part,the pair“2”is expounded and LW solver is revisited.The generalized Riemann problem(GRP)solver,as the discontinuous and nonlinear version of LW flow solver,and the gas kinetic scheme(GKS)solver,the microscopic LW solver,are all reviewed.The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed.Besides,the computational performance and prospective discussions are presented.

H^2-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm. To this end, an explicit Lyapunov function as a weighted and squared H2-norm of a small perturbation of the stationary solution is constructed. The authors show that by a suitable choice of the boundary feedback conditions, the H2- exponential stability of the stationary solution follows. Due to this fact, the system is stabilized over an infinite time interval. Furthermore, exponential estimates for the C norm are derived.

Waves Induced by the Appearance of Single-Point Heat Source in Constant Flow

Heating or cooling one-dimensional inviscid compressible flow can be modeled by the Euler equations with energy sources.A tricky situation is the sudden appearance of a single-point energy source term.This source is discontinuous in both the time and space directions,and results in multiple discontinuous waves in the solution.We establish a mathematical model of the generalized Riemann problem of the Euler equations with source term.Based on the double CRPs coupling method proposed by the authors,we determine the wave patterns of the solution.Theoretically,we prove the existence and uniqueness of solutions to both"heat removal"problem and"heat addition"problem.Our results provide a theoretical explanation for the effect of instantaneous addition or removal of heat on the fluid.