Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

Finite-time consensus of multi-agent systems driven by hyperbolic partial differential equations via boundary control

The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.

Oscillation of Nonlinear Impulsive Delay Hyperbolic Partial Differential Equations

In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.

A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations

We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

Dirichlet-to-Neumann Map for a Hyperbolic Equation

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.

OSCILLATION OF SOLUTIONS OF THE SYSTEMS OF QUASILINEAR HYPERBOLIC EQUATION UNDER NONLINEAR BOUNDARY CONDITION

Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.

Interaction of Conormal Waves With Strong and Weak Singularities For Semi－Linear Equations

We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.

OSCILLATION OF IMPULSIVE HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH DELAY

In this paper, oscillation properties of the solutions of impulsive hyperbolic equation with delay are investigated via the method of differential inequalities. Sufficient conditions for oscillations of the solutions are established.

The Nonlocal Singularly Perturbed Problems forNonlinear Hyperbolic Differential Equation

A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.

Evaluation of numerical schemes for capturing shock waves in modeling proppant transport in fractures

In petroleum engineering, the transport phenomenon of proppants in a fracture caused by hydraulic fracturing is captured by hyperbolic partial differential equations(PDEs). The solution of this kind of PDEs may encounter smooth transitions, or there can be large gradients of the field variables. The numerical challenge posed in a shock situation is that high-order finite difference schemes lead to significant oscillations in the vicinity of shocks despite that such schemes result in higher accuracy in smooth regions. On the other hand, first-order methods provide monotonic solution convergences near the shocks,while giving poorer accuracy in the smooth regions.Accurate numerical simulation of such systems is a challenging task using conventional numerical methods. In this paper, we investigate several shock-capturing schemes.The competency of each scheme was tested against onedimensional benchmark problems as well as published numerical experiments. The numerical results have shown good performance of high-resolution finite volume methods in capturing shocks by resolving discontinuities while maintaining accuracy in the smooth regions. Thesemethods along with Godunov splitting are applied to model proppant transport in fractures. It is concluded that the proposed scheme produces non-oscillatory and accurate results in obtaining a solution for proppant transport problems.

Homogenization of Hyperbolic Damped Stochastic Wave Equations

For a family of linear hyperbolic damped stochastic wave equations with rapidly oscillating coefficients, we establish the homogenization result by using the sigma-convergence method. This is achieved under an abstract assumption covering special cases like the periodicity, the almost periodicity and some others.

A CLASS OF NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR NONLINEAR HYPERBOLIC DIFFERENTIAL EQUATION

The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.

Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions

In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad’s moment equations or fail to include higher moments in apractical way like the entropy maximization approach. The new closure is based onPearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allowanisotropies and skewness in non-equilibrium. The closure relations are essentiallyexplicit and easy to evaluate. Hyperbolicity is shown numerically for a large range ofvalues. Numerical solutions of Riemann problems demonstrate the capability of thenew equations to handle strong non-equilibrium.

BOUNDARY DIFFERENCE-INTEGRAL EQUATION METHOD AND ITS ERROR ESTIMATES FOR SECOND ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain in R~3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.

Concentration Wave for a Class of Reaction Chromatography System with Pulse Injections

By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.

OSCILLATION OF NEUTRAL HYPERBOLIC DIFFERNTIAL EQUATION

In this paper, a neutral hyperbolic differential equation is studied, and some oscillation results are obtained.

Difference Scheme for Hyperbolic Heat Conduction Equation with Pulsed Heating Boundary

In this paper, the authors have analyzed the second-order hyperbolic differential equation with pulsed heating boundary, and tried to solve this kind of equation with numerical method. After analyzing the error of the difference scheme and comparing the numerical results with the theoretical results, the authors present some examples to show how the thermal wave propagates in materials. By analyzing the calculation results, the conditions to observe the thermal wave phenomena in the laboratory are conferred. Finally the heat transfer in a complex combined structure is calculated with this method. The result is quite different from the calculated result from the parabolic heat conduction equation.

On Some New Integral Inequalities for Functions in One and Two Variables

In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential equations under the integral inequalities.