Hyperbolic Relaxation Research Articles

On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations

Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However, in the intermediate regime (say, $\varepsilon=O(\Delta t)$), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptotic or stability analysis). In this work, we prove the uniform accuracy -- an optimal {\it a priori} error bound -- of a class of IMEX multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for linear hyperbolic systems with stiff relaxation. The proof is based on the energy estimate with a new multiplier technique. For nonlinear hyperbolic and kinetic equations, we numerically verify the same property using a series of examples.

The stability of poro-elastic wave equations in saturated porous media

Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the mathematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the first-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the first time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave field in reservoirs with pore/fracture networks, which is of great significance for unconventional oil and gas exploration.

On the 2D Cahn-Hilliard/Allen-Cahn equation with the inertial term

In this paper, we consider the initial boundary value problem for the hyperbolic relaxation of the 2D Cahn-Hilliard/Allen-Cahn equation with cubic nonlinearity of class C2. We prove that the semigroup generated by the weak solutions of this problem possesses a global attractor. We also improve the previously obtained results concerning the global attractors for the 2D Cahn-Hilliard equation with the inertial term.

A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary

We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for the relaxation system independently of the stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved using energy estimates and the Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (J. Differ. Equ. 167 (2000) 388–437).

Construction of boundary conditions for hyperbolic relaxation approximations I: The linearized Suliciu model

Starting with this paper, we intend to develop a program aiming at construction of boundary conditions (BCs) for hyperbolic relaxation systems. Physically, such BCs are not always available. The construction is based on the assumption that the relaxation systems and well-posed BCs for the corresponding equilibrium systems are given. This paper focuses on the linearized Suliciu model. We obtain strictly dissipative and compatible BCs for the linearized model with different non-characteristic boundaries. Moreover, the effectiveness of the constructed BCs is shown by resorting to formal asymptotic solutions and energy estimates.

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self‐similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.

Spectral stability of hydraulic shock profiles

By reduction to a generalized Sturm–Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia–Zumbrun and Yang–Zumbrun, this yields linear and nonlinear H2∩L1→H2 stability with sharp rates of decay in Lp, p≥2, the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.

Study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

In this paper, we study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms, with initial and homogenous Dirichlet boundary condition.

Existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn–Hilliard Equation

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn-Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an "approximately invariant manifold" $\mathcal{M}_0$ for such boundary value problem, that is we construct a narrow channel containing $\mathcal{M}_0$ and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a "transition layer structure" and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn-Hilliard equation is also performed.

Long-time dynamics of the Swift-Hohenberg equations

We consider the initial boundary value problems for the Swift-Hohenberg equation and its hyperbolic relaxation. Under the optimal conditions on the nonlinearity, we prove the well-posedness of the both problems and uniform (with respect to the initial data) global boundedness of the solutions. Then, we show that the both problems possess the global attractors of optimal regularity.

Stability of Hydraulic Shock Profiles

We establish nonlinear $$H^2\cap L^1 \rightarrow H^2$$ stability with sharp rates of decay in $$L^p$$, $$p\ge 2$$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans–Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.

An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by Sainte-Marie et al. in [1]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE.We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with a posteriori subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm.

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams–Moulton and Adams–Bashforth methods, whereas BDF methods preserve high-order accuracy. Subsequently we extend these results to semi-Lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings.

Boundary stabilization of hyperbolic balance laws with characteristic boundaries

This article is concerned with boundary stabilization of one-dimensional linear hyperbolic systems with characteristic boundaries. We assume that the systems satisfy a physically relevant structural stability condition for hyperbolic relaxation problems, which describe various non-equilibrium phenomena. By introducing new and simple Lyapunov functions, the structural stability condition is used to derive stabilization results for problems with characteristic boundaries. The result is illustrated with an application to the transport of neurofilaments in axons—a phenomenon studied in neuroscience.

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.Bibliography: 38 titles.

Hyperbolic relaxation method for elliptic equations

We show how the basic idea of parabolic Jacobi relaxation can be modified to obtain a new class of hyperbolic relaxation schemes that are suitable for the solution of elliptic equations. Some of the analytic and numerical properties of hyperbolic relaxation are examined. We describe its implementation as a first order system in a pseudospectral evolution code, demonstrating that certain elliptic equations can be solved within a framework for hyperbolic evolution systems. Applications include various initial data problems in numerical general relativity. In particular we generate initial data for the evolution of a massless scalar field, a single neutron star, and binary neutron star systems.

Linear, second order and unconditionally energy stable schemes for the viscous Cahn–Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method

In this paper, we consider numerical approximations for the viscous Cahn–Hilliard equation with hyperbolic relaxation. This type of equations processes energy-dissipative structure. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed “Invariant Energy Quadratization” approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes.

New structural conditions on decay property with regularity-loss for symmetric hyperbolic systems with non-symmetric relaxation

This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435–457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235–292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266] to cover those complicated models.