Search

This work revisits a recent finding by the first author concerning the local convergence of a regularized scalar conservation law. We significantly improve the original statement by establishing a global convergence result within the Lebesgue spaces L loc ∞ (R + ; L p (R)) , for any p ∈ [ 1 , ∞) , as the regularization parameter ℓ approaches zero. Notably, we demonstrate that this stability result is accompanied by a quantifiable rate of convergence. A key insight in our proof lies in the observation that the fluctuations of the solutions remain under control in low regularity spaces, allowing for a potential quantification of their behavior in the limit as ℓ → 0 . This is achieved through a careful asymptotic analysis of the perturbative terms in the regularized equation, which, in our view, constitutes a pivotal contribution to the core findings of this paper. [ABSTRACT FROM AUTHOR]

The work of Jia et al [[25]] discusses the stable control of one class of chaotic systems, and a control method based on the accurate exponential solution of a differential equation is used. Many problems in the broad spectrum of science require the solution of a partial differential equation. In [[23]], Karmran et al. propose a localized transform-based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo's sense with a weakly singular kernel. 10.3390/math8111854 25 Jia H., Guo C. The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System. [Extracted from the article]

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms. [ABSTRACT FROM AUTHOR]

The weighted essentially nonoscillatory (WENO) schemes are popular high-order numerical methods for hyperbolic conservation laws. When dealing with hyperbolic systems, WENO schemes are usually used in cooperation with the local characteristic decomposition, as the componentwise WENO reconstruction/interpolation procedure often produces oscillatory approximations near shocks. In this paper, we investigate local characteristic decomposition-free WENO schemes for a special class of hyperbolic systems endowed with a coordinate system of Riemann invariants. We apply the WENO procedure to the coordinate system of Riemann invariants instead of the local characteristic fields to save the expensive computational cost on local characteristic decomposition but meanwhile maintain the essentially nonoscillatory performance. Due to the nonlinear algebraic relation between the Riemann invariants and conserved variables, it is difficult to obtain the cell averages of Riemann invariants directly from those of conserved variables and vice versa; thus, we do not use the finite volume WENO schemes in this work. The same difficulty is also faced in the traditional Shu-Osher lemma [C.-W. Shu and S. Osher, J. Comput. Phys., 83 (1989), pp. 32-78]-based finite difference schemes, as the computation of fluxes is based on reconstruction as well. Therefore, we adopt the alternative formulation of the finite difference WENO scheme [Y. Jiang, C.-W. Shu, and M. Zhang, SIAM J. Sci. Comput., 35 (2013), pp. A1137-A1160, C.-W. Shu and S. Osher, J. Comput. Phys., 77 (1988), pp. 439-471] in this paper, which is based on interpolation for nodal values. The efficiency and good performance of our method are demonstrated by extensive numerical tests which indicate that the coordinate system of Riemann invariants is a good alternative of local characteristic fields for the WENO procedure. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem. [ABSTRACT FROM AUTHOR]

In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem. [ABSTRACT FROM AUTHOR]

The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them. [ABSTRACT FROM AUTHOR]

Copyright of Journal of University of Anbar for Pure Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, a regularized long wave travelling along an unsmooth boundary is depicted by the fractal calculus, and its fractal variational principle is established via the fractal semi-inverse method, which is very helpful to construct the conservation laws in the fractal space and to study the structure of the analytical solution, and a fractal solitary wave solution is obtained. [ABSTRACT FROM AUTHOR]

Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension d and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties. [ABSTRACT FROM AUTHOR]

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided. [ABSTRACT FROM AUTHOR]

This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal intervehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill-- Richards--Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter. [ABSTRACT FROM AUTHOR]

In this paper, we define the ruled surface in terms of the Darboux vector field of an isotropic curve in a complex 3-space, and we study the ruled surface as an elastic strip along the isotropic curve. First of all, we show that elastic strips along isotropic curves that serve critical points of the modified Sadowsky functional are characterized by three Euler-Lagrange equations. As a result, we give two conservation laws to characterize elastic strips of isotropic curves in complex 3-space and explain an isotropic helix in terms of the force vector. Finally, we give some examples to illustrate elastic strips with isotropic curves in a complex 3-space. [ABSTRACT FROM AUTHOR]

In this paper, we present two sufficient conditions keeping the total energy and cross-helicity conservation in spaces \underline {B}^{\alpha }_{3,VMO}(\Omega) and B^{\beta }_{3,\infty }(\Omega) for the ideal magnetohydrodynamic equations in a bounded domain. This generalizes the sufficient criteria for conserved quantities recently obtained by Bardos, Gwiazda, Świerczewska Gwiazda, Titi and Wiedemann [Proc. A. 475 (2019), 18 pp.] and by Wang and Zuo [J. Differential Equations 268 (2020), pp. 4079–4101]. [ABSTRACT FROM AUTHOR]

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any T 4 configurations. Here, we continue that study by showing that the differential inclusion does not contain T 5 configurations. [ABSTRACT FROM AUTHOR]

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem. [ABSTRACT FROM AUTHOR]

By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton's iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both L 2 - and L ∞ -error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our findings. [ABSTRACT FROM AUTHOR]

This paper describes conservation laws in general relativity (GR) dating back to the mass-energy conservation of Bondi and Sachs in the early 1960s but using 2-spinor techniques. The notion of conformal infinity is employed, and the highly original ideas of E. T. Newman are discussed in relation to twistor theory. The controversial NP constants are introduced, and their meaning is considered in a new light related to the problem of equations of motion in GR. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new type of Runge–Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple boundary treatment. The key idea is to hybridize two different spatial operators in an explicit RK scheme, utilizing local projected derivatives for inner RK stages and the usual DG spatial discretization for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. We also explore the connections between our method and Lax–Wendroff DG schemes and ADER-DG schemes. Numerical examples are given to confirm that the new RKDG method is as accurate as the original RKDG method, while being more compact, for problems including two-dimensional Euler equations for compressible gas dynamics. [ABSTRACT FROM AUTHOR]

Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [T. Chen and C.-W. Shu, J. Comput. Phys., 345 (2017), pp. 427-461; J. Chan, J. Comput. Phys., 362 (2018), pp. 346-374; T. Chen and C.-W. Shu, CSIAM Trans. Appl. Math., 1 (2020), pp. 1-52] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299-1324; Y. Liu, J. Lu, and C.-W. Shu, SIAM J. Sci. Comput., 44 (2022), pp. A230-A259], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $ \ell>0 $ and conserves an $ H^1 $ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $ \ell $ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization.This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case. [ABSTRACT FROM AUTHOR]

Cell average decomposition (CAD) plays a critical role in constructing boundpreserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant--Friedrichs--Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss--Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D P2 and P3 spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional P2 and P3 spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carath\'eodory's theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D Pk spaces and general 2D Qk spaces of an arbitrary k\geq 1. For the widely used 2D Pk spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the orbital stability of periodic peaked traveling waves (peakons) for a nonlinear quartic μ -Camassa–Holm equation. The equation is a μ -version of the nonlinear quartic Camassa–Holm equation which was proposed by Anco and Recio (J Phys A Math Theor 52:125–203, 2019). The equation admits the periodic peakons. It is shown that the periodic peakons are orbitally stable under small perturbations in the energy space by finding inequalities related to the three conservation laws with global maximum and minimum of the solution. [ABSTRACT FROM AUTHOR]

The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the (G ′ / G 2) − expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the (G ′ / G 2) − expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem. The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem. [ABSTRACT FROM AUTHOR]

In this paper we shall establish an a priori L²-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of onedimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space is achieved for many widely used numerical fluxes, no matter whether the solution contains sonic points or not. The main tools used in this paper are the matrix transferring process and the generalized Gauss--Radau projection of the reference functions, depending on the relative upwind effect. Finally some numerical experiments are given to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simple, purely stochastic model characterized by two conserved quantities (mass density a and energy density h) which is known to display a condensation transition when h > 2 a 2 : in the localized phase a single site hosts a finite fraction of the whole energy. Its equilibrium properties in the thermodynamic limit are known and in a recent paper [G. Gotti, S. Iubini and P. Politi, Finite-size localization scenarios in condensation transitions, Phys. Rev. E 103 (2021) 052133] we studied the transition for finite systems. Here, we analyze finite-size effects on the energy distribution and on the relaxation dynamics, showing that extremely large systems should be studied in order to observe the asymptotic distribution and even larger systems should be simulated in order to observe the expected relaxation dynamics. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of hyperbolic systems of conservation laws admitting delta-waves. The Riemann problem is completely solved. The solutions exhibit four kinds of wave patterns: the first contains a vacuum state bounded by two contact discontinuities; the second involves the overcompressible delta-wave; the third and fourth include a vacuum state bounded by a contact delta-wave and a contact discontinuity. The highlight of this paper is the contact delta-waves. [ABSTRACT FROM AUTHOR]

A key consideration in the development of numerical schemes for time-dependent partial differential equations (PDEs) is the ability to preserve certain properties of the continuum solution, such as associated conservation laws or other geometric structures of the solution. There is a long history of the development and analysis of such structure-preserving discretization schemes, including both proofs that standard schemes have structure-preserving properties and proposals for novel schemes that achieve both high-order accuracy and exact preservation of certain properties of the continuum differential equation. When coupled with implicit time-stepping methods, a major downside to these schemes is that their structure-preserving properties generally rely on an exact solution of the (possibly nonlinear) systems of equations defining each time step in the discrete scheme. For small systems, this is often possible (up to the accuracy of floating-point arithmetic), but it becomes impractical for the large linear systems that arise when considering typical discretization of space-time PDEs. In this paper, we propose a modification to the standard flexible generalized minimum residual iteration that enforces selected constraints on approximate numerical solutions. We demonstrate its application to both systems of conservation laws and dissipative systems. [ABSTRACT FROM AUTHOR]

In a recent series of papers by Lou et al it was conjectured that higher dimensional integrable equations may be constructed by utilizing some conservation laws of (1 + 1) -dimensional systems. We prove that the deformation algorithm introduced in (Lou et al 2023 J. High Energy Phys. 2023 018), applied to Lax integrable (1 + 1) -dimensional systems, produces Lax integrable higher dimensional systems. The same property is enjoyed by the generalized deformation algorithm introduced in (Lou et al 2023 Chin. Phys. Lett. 40 020201); we present a novel example of a (2 + 1) -dimensional deformation of KdV equation obtained by generalized deformation. The deformed systems obtained by such procedure, however, pose a serious challenge because most of the mathematical structures that the (1 + 1) -dimensional systems possess are lost. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Riemann-type hydrodynamical equation and establish the existence of a weak solution to the equation in a lower order Sobolev space (Hs(ℝ))N with 1 < s ≤ 3/2 by using a modified pseudo-parabolic regularization method. Although the absence of some important conservation laws leads to the failure of global existence of the approximate solutions, it has been addressed by making good use of the structure of the system. [ABSTRACT FROM AUTHOR]

In this paper, a class of strictly hyperbolic systems of conservation laws which arises in connection with enhanced oil recovery is studied. The Riemann problem is solved analytically. The Riemann solutions with two kinds of different structures involving the delta-shock are obtained. For delta-shock, the generalized Rankine–Hugoniot relations and over-compressive delta-entropy condition are clarified. Further, the existence and uniqueness of delta-shock are established. The theoretical analysis is tested accurately by the numerical results. [ABSTRACT FROM AUTHOR]

This paper put forward a new fully discrete scheme construction method -- double entropy condition solution formula method. With that, we turn the state-of-the-art semi-discrete WENO+RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work, the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper studies a Cauchy problem for a scalar multidimensional (multi-d) viscous convex conservation law, in which the initial data is a planar viscous shock with a multi-d periodic perturbation. We show that if the wave-strength and perturbation are both small, then the viscous shock is stable in the L∞ (ℝn) space with an exponential decay rate. One contribution of the paper is to establish a new framework to study the stability of viscous shocks in multiple dimensions, where the elementary energy method with the antiderivative technique can be used. The idea is to decompose the multi-d perturbation into a one-dimensional function and a multi-d remainder, where the former can well define its antiderivative and the latter satisfies the Poincaré inequality over a periodic domain. [ABSTRACT FROM AUTHOR]

The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost. [ABSTRACT FROM AUTHOR]

This paper deals with a problem that we found when directly implementing the scalar MUSCL scheme to spectral coefficient equations of the gPC-Galerkin method. The order degradation at extrema of spectral coefficients and non-uniform reconstruction process lead to a distortion of standard deviation at some extrema, especially where the peak consists of several extrema of coefficients. From the perspective of probability space, a spectral MUSCL scheme is proposed based on the generalized minmod limiter to avoid the defects mentioned above when applying the scalar MUSCL. In this paper, we present some properties of the new scheme related to total variation and demonstrate them by the uncertain linear scalar conservation law. Finally, we employ this scheme to compressible Euler equations, and a good correlation of standard deviation is obtained. The details of implementation of the gPC-Galerkin method for Euler equations are also discussed, and the approximate Jacobian matrix is adopted. [ABSTRACT FROM AUTHOR]

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

In present paper, we mainly consider the wave-breaking phenomena and persistence properties of a nonlinear dissipative Camassa–Holm equation. The precise blow-up criteria of the Cauchy problem are established and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Due to the presence of high order nonlinear term, the conservation law E = ∫ R u 2 + u x 2 d x is losed. This difficulty has been addressed by establishing the energy inequality. Finally, we establish the persistence properties and some unique continuation properties of the solutions in the weighted L p -spaces L ϕ p (R) := L p (R , ϕ p d x). [ABSTRACT FROM AUTHOR]

This paper is focused on the Riemann problem for a 2 × 2 hyperbolic system of conservation laws with a time-gradually-degenerate damping. Two kinds of non-self-similar solutions involving the delta-shocks and vacuum are obtained using the variable substitution method. The generalized Rankine-Hugoniot relation and entropy condition are clarified for the delta-shock. Furthermore, the vanishing viscosity method proves the existence, uniqueness, and stability of non-self-similar solutions. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a more general discrete 2 × 2 matrix spectral problem. Starting from this spectral problem, the positive and negative integrable lattice hierarchies are constructed based on the Tu scheme, then by considering linear combination of the positive and negative lattice hierarchies, we give a more general integrable lattice hierarchy, which can reduce to the well-known Ablowitz–Ladik lattice and the discrete modified Korteweg–de Vries (mKdV) equation. In particular, we obtain some local and nonlocal integrable lattice equations, including reverse-space discrete mKdV equation, reverse-space complex discrete mKdV equation, higher-order discrete mKdV equation, higher-order complex discrete mKdV equation, higher-order reverse-space discrete mKdV equation and higher-order reverse-space complex discrete mKdV equation. In additional, infinitely many conservation laws and Darboux transformation (DT) for the first non-trivial system in the hierarchy are established with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained DT. The results in this paper might be helpful for understanding some physical phenomena. [ABSTRACT FROM AUTHOR]

In this paper, in order to provides a powerful new tool for quantitative and qualitative analysis of dynamics properties in flexible mechanical multibody systems, the symmetry theory and numerical algorithms for preserving structure in modern analytical mechanics is introduced into flexible multibody dynamics. First, taking the hub-beam systems as an example, the original nonlinear partial differential-integral equations of the system dynamics model are discretized into the finite-dimensional Lagrange equations by using the assumed modal method. Second, the group analysis theory is introduced and the criterion equations and the corresponding conserved quantities of Noether symmetries are given according to the invariance principle, which provide an effective way for analytic integral theory of dynamic equations. Finally, a conserved quantity-preserving numerical algorithm is constructed by coordinates incremental discrete gradient, which makes full use of the invariance of conserved quantity to eliminate the error consumption for a long time. The simulation results show that the deeper mechanical laws and motion characteristics of flexible mechanical multibody systems dynamics can be obtained with the help of symmetries and conserved quantities, which can provide reference for more precise dynamic optimization design and advanced control of systems. [ABSTRACT FROM AUTHOR]

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on the coarse time-mesh. Lagrange's linear interpolation formula is employed to obtain all preliminary solutions on the fine time-mesh. Based on the preliminary solutions, Taylor's formula is utilized to construct a linear system for the equation on the fine time-mesh. The convergence and stability of the scheme is analyzed, providing the convergence rates of O (τ F 2 + τ C 4 + h 4) in the discrete L ∞ -norm for u (x , t) and in the discrete L 2 -norm for ρ (x , t) . Numerical simulation results show that the proposed scheme achieves equivalent error levels and convergence rates to the nonlinear scheme, while also reducing CPU time by over half, which indicates that the new method is more efficient. Furthermore, compared to the earlier time two-mesh method developed by the authors, the proposed scheme significantly reduces the error between the numerical and exact solutions, which means that the proposed scheme is more accurate. Additionally, the effectiveness of the new scheme is discussed in terms of the corresponding conservation laws and long-time simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach. [ABSTRACT FROM AUTHOR]

The Herglotz variational principle offers an effective method for studying nonconservative system dynamics. The aim of this paper is to study the conservation laws of nonholonomic systems by using the Herglotz type generalized variational principle and establish Noether's theorem and its inverse theorem for this system. In deriving the equations of motion, we use the Suslov definition of the reciprocity relation between differential and variational operations. First, the Herglotz type generalized variational principle is listed, and the Herglotz type Chaplygin equations for nonconservative nonholonomic systems are deduced. Second, Noether's theorem and Noether's inverse theorem are established, and the Herglotz type conservation laws are given. Finally, an example is provided to illustrate the practical implementation of the findings. [ABSTRACT FROM AUTHOR]

It is interesting and challenging to study conservation laws with discontinuous source terms and explore how the delta shock wave is influenced by the discontinuous source term. However, so far, few results have been obtained about it. In this paper, the Riemann problem for the pressureless Euler equations with a discontinuous source term is considered. The delta shock wave solution is obtained by combining the generalized Rankine-Hugoniot conditions, together with the method of characteristics for various kinds of different situations, and the impact of the discontinuous source term on the delta shock front are precisely illustrated. Moreover, during the process of constructing the Riemann solution, some interesting phenomena are also observed, such as the disappearance of the delta shock wave and the occurrence of the vacuum state, etc. [ABSTRACT FROM AUTHOR]

This paper investigates the (3 + 1) -dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the (3 + 1) -dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie's continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver's standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov's generalized theorem and quasi-self-adjointness condition. [ABSTRACT FROM AUTHOR]