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Abstract This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1) τ ∂ t 2 u (x , t) − Δ u (x , t) + a (x) ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ C Σ × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ C Σ × (0 , T) , u (x , 0) = ε f (x) , x ∈ C Σ , τ ∂ t u (x , 0) = τ ε g (x) , x ∈ C Σ , where C Σ is a cone-like domain in R N (N ≥ 2) defined as C Σ = int r ω ∈ R N ; r ≥ 0 , ω ∈ Σ with a connected open set Σ in S N − 1 with smooth boundary ∂ Σ. If N = 1 , then we only consider two cases C Σ = (0 , ∞) and C Σ = R. Here a (x) is a non-zero coefficient of ∂ t u which could be complex-valued and space-dependent, λ ∈ ℂ is a fixed constant, and ε > 0 is a small parameter. The constants τ = 0 , 1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable. [ABSTRACT FROM AUTHOR]

In this paper we consider a hyperbolic Keller-Segel system with a logistic source in two dimension. We show the system has a global smooth solution upon small perturbation around a constant equilibrium and the solution satisfies a dissipative energy inequality. To do this we find a convex entropy functional and a compensating matrix, which transforms the partially dissipative system into a uniformly dissipative one. Those two ingredients were crucial for the study of a partially dissipative hyperbolic system (Hanouzet and Natalini in Arch. Ration. Mech. Anal. 169(2):89-117, 2003; Kawashima in Ph.D. Thesis, Kyoto University, 1983; Yong in Arch. Ration. Mech. Anal. 172(2):247-266, 2004). [ABSTRACT FROM AUTHOR]

In this study, a collocation approach is introduced to solve second-order two-dimensional hyperbolic telegraph equation under the initial and boundary conditions. The method is based on the Bessel functions of the first kind, matrix operations and collocation points. The method is constructed in four steps for the considered problem. In first step we construct the fundamental relations for the solution method. By using the collocation points and matrix operations, second step gives the constructing of the main matrix equation. In third step, matrix forms are created for the initial and boundary conditions. We compute the approximate solutions by combining second and third steps. Algorithm of the proposed method is given. Later, error estimation technique is presented and the approximate solutions are improved. Numerical applications are included to demonstrate the validity and applicability of the presented method. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation u t ⁢ t = L ⁢ u + f ⁢ ( x , t ) in ⁢ Ω T = Ω × ( 0 , T ) , u_{tt}=Lu+f(x,t)\quad\text{in }\Omega_{T}=\Omega\times(0,T), where L is a nonlinear operator and ϕ ⁢ ( x , t , ⋅ ) {\phi(x,t,\cdot\,)} , f ⁢ ( x , t ) {f(x,t)} and the exponents of the nonlinearities p ⁢ ( x , t ) {p(x,t)} and μ ⁢ ( x , t ) {\mu(x,t)} are given functions. [ABSTRACT FROM AUTHOR]

We present a new scale $$\mathcal {U}^{t,s}_p$$ ( $$s<-t<0$$ and $$1\le p <\infty $$ ) of anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer operator $$\mathcal {L}_g \varphi = (g \cdot \varphi ) \circ T^{-1}$$ associated to a hyperbolic dynamical system T has good spectral properties. When $$p=1$$ and t is an integer, the spaces are analogous to the 'geometric' spaces $$\mathcal {B}^{t,|s+t|}$$ considered by Gouëzel and Liverani (Ergod Theory Dyn Syst 26:189-217, 2006). When $$p>1$$ and $$-1+1/p

The Gauss–Codazzi system for the given metric whose Gauss curvature is negative, is considered in the present paper. We provide some suitable conditions on the given metric such that the system admits a global BV entropy solution. Our main approach is applying the method of seeking global BV entropy solutions to the hyperbolic system of balance laws to the Gauss–Codazzi system. The global existence of BV solutions to the Gauss–Codazzi system yields a C 1 , 1 isometric immersion of two dimensional manifold into R 3 . [ABSTRACT FROM AUTHOR]

The paper concerns nonlocal time-periodic boundary value problems for first-order integro-differential hyperbolic systems with boundary inputs. The systems are subjected to integral boundary conditions. Under natural regularity assumptions on the data, it is shown that the problems display completely non-resonant behavior and satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First, using the Cattaneo's relaxation approach, we reformulate the system of governing equations for the pollutant transport by shallow water flows over non-flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms. The proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection–diffusion models. This turns out to give a larger stability range for the choice of the time step. Second, following a flux splitting approach, we derive a novel numerical method to discretise the resulting problem. In particular, we propose a new flux splitting and study the associated two systems of differential equations, called the “hydrodynamic” and the “relaxed diffusive” system, respectively. For the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the Godunov-type. These schemes are simple to implement, robust, accurate and fast when compared with existing methods. The resulting method is implemented on unstructured meshes and is systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems including non-flat topography and wetting and drying problems. Formal second order accuracy is assessed through convergence rates studies. [ABSTRACT FROM AUTHOR]

Simple waves and double waves as exact solutions of quasilinear hyperbolic equations in hydrodynamics may blow up, and, so, generally, there are points at which these solutions are violated at some definite instants of time. Finding the first instant of time for the violation of a simple wave solution is a known method, while this problem for the double waves is not a well defined and an easy task. Here, a two and a three dimensional double wave solutions are reviewed for some hydrodynamical systems to show that, for the 2-D case for some special solutions, one can find a local minimum for the critical time, while for some other solutions not. For the 3-D case, generically, there is no local minimum for the critical time, but at very special situations, the local minimum critical times may exist. Moreover, the dynamics of critical curves and points are briefly discussed. [ABSTRACT FROM AUTHOR]

In the current paper, we derive the a priori error analysis ( hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature. [ABSTRACT FROM AUTHOR]

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity. Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency. The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model. The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary. [ABSTRACT FROM AUTHOR]

In this article the authors continue the discussion in [9] about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincaré operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree. [ABSTRACT FROM AUTHOR]

In this paper the correct solvability of mixed problem for linear hyperbolic countable system of first order equations with two independent variables is proved. [ABSTRACT FROM AUTHOR]

Using the hyperbolic heat conduction equation found from the condition of heat flow relaxation and the temperature gradient in the formula of the Fourier law, an exact analytical solution of the boundary problem of dynamic thermoelasticity for an infinite plate with symmetric boundary conditions of the first kind is obtained. It is shown that stresses change discontinuously in time with a periodic change in their sign at every point of the space. Under a sustained character, the process of changes in stresses occurs in the form of a string fixed at both ends and having kinks (stress jumps) moving along the spatial variable in time. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144-8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235-256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundström, Math. Comput. 26 (1972) 649-686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems. [ABSTRACT FROM AUTHOR]

In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In many application domains, the preferred approaches to the numerical solution of hyperbolic partial differential equations such as conservation laws are formulated as finite difference schemes. While finite difference schemes are amenable to physical interpretation, one disadvantage of finite difference formulations is that it is relatively difficult to derive the so-called goal oriented a posteriori error estimates. A posteriori error estimates provide a computational approach to numerically compute accurate estimates in the error in specified quantities computed from a numerical solution. Widely used for finite element approximations, a posteriori error estimates yield substantial benefits in terms of quantifying reliability of numerical simulations and efficient adaptive error control. The chief difficulties in formulating a posteriori error estimates for finite difference schemes is introducing a variational formulation–and the associated adjoint problem–and a systematic definition of residual errors. In this paper, we approach this problem by first deriving an equivalency between a finite element method and the Lax–Wendroff finite volume method. We then obtain an adjoint based error representation formula for solutions obtained with this method. Results from linear and nonlinear viscous conservation laws are given. [Copyright &y& Elsevier]

A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper. [ABSTRACT FROM AUTHOR]

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously inL2- andH1-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes. [ABSTRACT FROM AUTHOR]

The article presents an implementation of the energy inequality method for hyperbolic operator-differential equations with variable domains. This method was applied to prove the Hadamard well-posedness of a boundary value problem. The well-posedness of the Cauchy problem for second order hyperbolic operator-differential equations is proved using weak differentiation, and the energy inequality for strong solutions is proved using formula for the weak operator derivative.

Abstract: In this paper, we propose a new high order three-level implicit method based on off-step discretization on a non-uniform mesh for the solution of 1-D non-linear hyperbolic partial differential equation of the form utt = uxx + g(x, t, u, ux , ut ), subject to appropriate initial and Dirichlet boundary conditions. We use only three evaluations of the function g and three grid points at each time level in a compact cell. Our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation. The method is convergent for uniform mesh. Numerical results are provided to justify the usefulness of the proposed method. [Copyright &y& Elsevier]

A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suitable for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples. [ABSTRACT FROM AUTHOR]

In this article, we study oscillatory properties of solutions to neutral nonlinear impulsive hyperbolic partial differential equations with several delays. We establish sufficient conditions for oscillation of all solutions. [ABSTRACT FROM AUTHOR]

In this work, we study a boundary problem with non-local conditions, by relating values of the unknown function with various characteristics. The parabolic-hyperbolic equation with three lines of type changing is equivalently reduced to a system of Volterra integral equations of the second kind. [ABSTRACT FROM AUTHOR]

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures and small Seifert fibred structures on closed 3-manifolds. [ABSTRACT FROM AUTHOR]

Abstract: In this work we study the nonlinear sinh–Gordon equation in (1+1), (2+1) and (3+1) dimensions. We use the simplified form of Hirota’s method, to derive one and two soliton solutions for each equation. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of solutions under the Pettis integrability assumption for an initial value problem for a hyperbolic differential inclusion of Caputo fractional order. We apply the measure of weak noncompactness technique. [Copyright &y& Elsevier]

Abstract: In this paper, based on the F-expansion method and the extended version of F-expansion method we investigate the exact solutions of the sine–Gordon equation and sinh-Gordon equation, respectively. Firstly, several special transformations are introduced to change the sine–Gordon and sinh-Gordon equations into polynomial nonlinear evolution equations. Then these polynomial nonlinear evolution equations are solved by the F-expansion method and its extended version. Finally, more new exact Jacobian elliptic function solutions and soliton solutions are constructed explicitly. [Copyright &y& Elsevier]

Abstract: We consider quasilinear autonomous inclusions of hyperbolic type. The dynamics of all weak solutions defined on the positive semi-axis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. The classes of mathematical models for piezoelectric fields containing the multidimensional law are studied. The conditions of the output of each weak solution for this problem at stationary states are given. We consider as a particular case the piezoelectric model for PZT-4 piezoceramics as one of the possible applications. [Copyright &y& Elsevier]

Abstract: In this paper, we present the development of new explicit group relaxation methods which solve the two dimensional second order hyperbolic telegraph equation subject to specific initial and Dirichlet boundary conditions. The explicit group methods use small fixed group formulations derived from a combination of the rotated five-point finite difference approximation together with the centered five-point centered difference approximation on different grid spacings. The resulting schemes involve three levels finite difference approximations with second order accuracies. Analyses are presented to confirm the unconditional stability of the difference schemes. Numerical experimentations are also conducted to compare the new methods with some existing schemes. [Copyright &y& Elsevier]

Abstract: We consider degenerate Kirchhoff equations with a small parameter ε in front of the second-order time-derivative. It is well known that these equations admit global solutions when ε is small enough, and that these solutions decay as with the same rate of solutions of the limit problem (of parabolic type). In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as , and as . Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as ). Proofs involve a nonlinear step where we separate Fourier components with respect to the lowest frequency, followed by a linear step where we exploit weighted versions of classical energies. [Copyright &y& Elsevier]

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension n. Furthermore, in the onedimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous. [ABSTRACT FROM AUTHOR]

If the stable, centre and unstable foliations of a partially hyperbolic system are quasi-isometric, the system has Global Product Structure. This result also applies to Anosov systems and to other invariant splittings. If a partially hyperbolic system on a manifold with an abelian fundamental group has quasi-isometric stable and unstable foliations, the centre foliation is without holonomy. If, further, the system has Global Product Structure, then all centre leaves are homeomorphic. [ABSTRACT FROM AUTHOR]

Abstract: This work is a continuation of our previous work [Z.Q. Shao, Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data, Nonlinear Anal. Real World Appl. 11 (2010) 3791–3808]. In the present paper we investigate the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution only contains the shocks and contact discontinuities, and at least a shock wave. The perturbations are in BV but they are assumed to be -smooth, with bounded and possibly large -norms. We get a lower bound of the lifespan of the piecewise solution to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to one-dimensional Euler equations of gas dynamics for a compressible, inviscid, non-heat conducting gas in Eulerian coordinates, are also given. [Copyright &y& Elsevier]

By means of the classical symmetry method, a hyperbolic Monge-Ampère equation is investigated. The symmetry group is studied and its corresponding group invariant solutions are constructed. Based on the associated vector of the obtained symmetry, the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampère equation, from which two interesting classes of solutions to the hyperbolic Monge-Ampère equation are obtained successfully. [ABSTRACT FROM AUTHOR]

Abstract: For 1-D quasilinear hyperbolic systems, the strict dissipation or the weak linear degeneracy can prevent the formation of singularity. More precisely, if all the inhomogeneous sources are strictly dissipative, or all the characteristics are weakly linearly degenerate and the system is homogeneous, then the Cauchy problem with small and decaying initial data admits a unique global classical solution. In this paper, under some suitable hypotheses on the interaction, new kinds of weighted formulas of wave decomposition are developed to show the same result for a general class of combined systems, in which a part of equations possesses the strict dissipation and the others are weakly linearly degenerate. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to study the long time behavior of the energy of solutions to 2 by 2 linear hyperbolic systems. Some blow-up results at are given for a large class of initial data of the Cauchy problem. We shall prove the optimality of these results for a special class of systems. We present also an example of generalized energy conservation under a stabilization condition. [Copyright &y& Elsevier]

Abstract: This work presents a numerical procedure for obtaining approximated solutions for one-dimensional fluid–structure interaction (FSI) models, which are used in transient analyses of liquid-filled piping systems. The FSI model considered herein is formed by a system of hyperbolic partial differential equations and describes, simultaneously, pressure waves propagating in the liquid as well as axial, shear and bending waves traveling in the pipe walls. By taking advantage of an operator splitting technique, the flux term is split away from the source one, giving rise to a sequence of simpler problems formed by a set of homogeneous hyperbolic differential equations and by a set of ordinary differential equations in time. The numerical procedure is constructed by advancing in time sequentially through these sets of equations by employing Glimm''s method and Gear''s stiff method, respectively. To implement Glimm''s method, analytical solutions for the associated Riemann problems are presented. The boundary conditions are properly accounted for in Glimm''s method by formulating and analytically solving suitable (non-classical) Riemann problems for the pipe''s ends. The proposed numerical procedure is used to obtain numerical approximations for the well-known eight-equation FSI model for two closed piping systems, in which transients are generated by the impact of a rod onto one of the ends. The obtained numerical results are compared with experimental data available in the literature and very good agreement is found. [Copyright &y& Elsevier]

Based on the theory of semi-global C solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method. [ABSTRACT FROM AUTHOR]

We introduce non-standard, finite-difference schemes to approximate nonnegative solutions of a weakly hyperbolic (that is, a hyperbolic partial differential equation in which the second-order time-derivative is multiplied by a relatively small positive constant), nonlinear partial differential equation that generalizes the well-known equation of Fisher-KPP from mathematical biology. The methods are consistent of order O(Δ t+(Δ x)2). As a means to verify the validity of the techniques, we compare our numerical simulations with known exact solutions of particular cases of our model. The results show that there is an excellent agreement between the theory and the computational outcomes. [ABSTRACT FROM AUTHOR]

Sine-Gordon kinks are non-dispersive solutions in a much studied integrable system. Recent studies on sine-Gordon kinks with space-dependent square-well-type potentials have revealed interesting dynamics of a single kink interacting with wells and barriers. In this paper, we study a class of smooth space-dependent potentials and discuss the dynamics of one kink in the presence of different wells. We also present values for the critical velocity for different types of barriers. Furthermore, we study two kinks interacting with various wells and describe interesting trajectories such as double-trapping, kink knock-out and double-escape. [ABSTRACT FROM AUTHOR]

Abstract: We study a family of generalized slope limiters in two dimensions for Runge–Kutta discontinuous Galerkin (RKDG) solutions of advection-diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiter’s advantages and disadvantages. We then introduce a series of coupled p-enrichment schemes that may be used as standalone dynamic p-enrichment strategies, or may be augmented via any in the family of variable-in-p slope limiters presented. [Copyright &y& Elsevier]