Large Wave Speed Research Articles

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

Plane wave stability analysis of Hartree and quantum dissipative systems

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual nonlinear Schrödinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.

Traveling wave solutions for a two-species competitive Keller–Segel chemotaxis system

In this paper, the traveling wave problem for a two-species competition reaction–diffusion–chemotaxis Lotka–Volterra system is investigated. Upper and lower solutions method and fixed point theory are employed to show the existence of traveling wave solutions connecting the coexistence constant steady state with zero state for all large enough wave speed c, and conversely, when c is small, we prove there is no traveling wave solution.

Crustal Structure Constraints From the Detection of the SsPp Phase on Mars

AbstractThe shallowest intracrustal layer (extending to 8 ± 2 km depth) beneath the Mars InSight Lander site exhibits low seismic wave velocity, which is likely related to a combination of high porosity and other lithological factors. The SsPp phase, an SV‐ to P‐wave reflection on the receiver side, is naturally suited for constraining the seismic structure of this top crustal layer since its prominent signal makes it observable with a single station without the need for stacking. We have analyzed six broadband and low‐frequency seismic events recorded on Mars and made the first coherent detection of the SsPp phase on the red planet. The timing and amplitude of SsPp confirm the existence of the ∼8 km interface in the crust and the large wave speed (or impedance) contrast across it. With our new constraints from the SsPp phase, we determined that the average P‐wave speed in the top crustal layer is between 2.5 and 3.2 km/s, which is a more precise and robust estimate than the previous range of 2.0–3.5 km/s obtained by receiver function analysis. The low velocity of Layer 1 likely results from the presence of relatively low‐density lithified sedimentary rocks and/or aqueously altered igneous rocks that also have a significant amount of porosity, possibly as much as 22%–30% by volume (assuming an aspect ratio of 0.1 for the pore space). These porosities and average P‐wave speeds are compatible with our current understanding of the upper crustal stratigraphy beneath the InSight Lander site.

Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

We consider traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity. We investigate how the existence of traveling waves, their shapes, and asymptotic behavior change with the presence or absence of an inertial term. These are studied by applying the framework that combines Poincare compactification, classical dynamical systems theory, and geometric methods for the desingularization of vector fields. We report that the presence of this term causes the shapes to change significantly for sufficiently large wave speeds. For more information see https://ejde.math.txstate.edu/Volumes/2023/05/abstr.html

Stability of traveling wave solutions for a nonlocal Lotka-Volterra model.

In this paper, we studied the stability of traveling wave solutions of a two-species Lotka-Volterra competition model in the form of a coupled system of reaction diffusion equations with nonlocal intraspecific and interspecific competitions in space at times. First, the uniform upper bounds for the solutions of the model was proved. By using the anti-weighted method and the energy estimates, the asymptotic stability of traveling waves with large wave speeds of the system was established.

How do cell crowding and starvation affect avascular tumor growth of the EMT6/Ro tumor?

Here we re-examine experimental in vitro data for the EMT6/Ro tumor volume and viable rim thickness. This shows that the growth speed is constant in time, and independent of nutrient concentration at large concentrations, but that the viable rim thickness increases in this high concentration regime. We then present a simple mechanistic reaction-diffusion equation that includes crowding and starvation effects, and show that the model qualitatively captures the experimental observations. Moreover, the model predicts that the cancer cell concentration is characterised by a wave pulse (soliton), and the pulse shape is explored through zero’th order perturbation analysis corresponding to large wave speeds. It is shown that this zero’th order term is dominant for the experimental condition, further indicating that the non-linear reaction governs the pulse characteristic shape. Finally, at low nutrient concentrations we find that the front is a pulled-front, that is, the growth speed is determined by the cell multiplication at the wave front edge in accordance with earlier modelling approaches.

Evidence for crustal seismic anisotropy at the InSight lander site

We analyzed broadband and low-frequency events recorded on Mars and made the first detection of horizontally polarized shear wave reflections, which help constrain the crustal structure at NASA's InSight lander site. Coherent signals from five well-recorded marsquakes appear to be independent of the focal depth and are consistent with SH-wave reflections off the topmost crustal interface (8 ± 2 km). This phase confirms the existence of the ∼8 km interface in the crust and the large wave speed (or impedance) contrast across it. The range of acceptable parameters determined from the detected SH-wave reflections differs from the majority of the vertically polarized shear wave models resulting from a previous receiver function study, indicating that the velocity of the vertically polarized waves is larger than that of horizontally polarized waves. We propose that this inconsistency results from the presence of seismic anisotropy within the top crustal layer at the lander site. Modeling results show that dry- or liquid-filled cracks/fractures and igneous intrusions can reproduce the observed radial anisotropy.

On the stability of solitary wave solutions for a generalized fractional Benjamin–Bona–Mahony equation

In this paper we establish a rigorous spectral stability analysis for solitary waves associated to a generalized fractional Benjamin–Bona–Mahony type equation. Besides the well known smooth and positive solitary wave with large wave speed, we present the existence of smooth negative solitary waves having small wave speed. The spectral stability is then determined by analysing the behaviour of the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum at the solitary wave. Since the analytical solution is not known, we generate the negative solitary waves numerically by using Petviashvili method. We also present some numerical experiments to observe the stability properties of solitary waves for various values of the order of nonlinearity and fractional derivative. Some remarks concerning the orbital stability are also celebrated.

Traveling wave phenomena of a nonlocal reaction-diffusion equation with degenerate nonlinearity

This paper deals with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. We study the existence of traveling wave solutions which may be non-monotonic based on the two-point boundary value problem and Schauder’s fixed point theorem. We are excited to find that the unknown positive steady state is exactly a unique positive equilibrium for the large wave speed and the monotonicity of traveling waves depends on the wave speed. On the other hand, we demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions.

Stability of traveling waves of the nonlocal Fisher–KPP equation

The existence of traveling wave solutions of the nonlocal Fisher–KPP equation has been widely studied. In this paper we concern the stability of traveling wave solutions of the equation. By the anti-weighted method and the energy estimates, we establish the asymptotic stability of traveling wave solutions with large wave speeds.

Body Wave Speed Structure of Eastern North America

AbstractThe eastern, passive margin of the North American continent is more heterogeneous than originally envisioned. Imaged by teleseismic body wave arrivals, both compressional Vp and shear Vsh, the lithospheric mantle of the stable, eastern North America contains rapid and large wave speed variations from the fast, older interior toward the slower, margin structures along the continental edge. At multiple regions at the continental edge, wave speeds indicate the presence of a small amount of melt, <1%–2%. Melt assessments were computed using a suite of mineralogical/petrological models to convert Vp and Vsh into absolute temperatures and melt fraction. Moreover, along the Atlantic margin, wave speeds are lower than expected for a previously rifted margin. Within the continent interior, wave speeds in the mantle lithosphere vary spatially at wavelengths, suggesting strong compositional and mineralogical influence likely a result of later stage alteration from volatile rich fluids.

Traveling wavefronts in a two-species chemotaxis model with Lotka–Volterra competitive kinetics

This paper is devoted to the study of traveling wavefronts for a class of two-species chemotaxis models with Lotka–Volterra competitive kinetics. The existence of traveling wavefronts of large wave speed is established by the perturbation method. Some numerical simulations are presented to illustrate our main results.

Large speed traveling waves for the Rosenzweig–MacArthur predator–prey model with spatial diffusion

This paper focuses on traveling wave solutions for the so-called Rosenzweig–MacArthur predator–prey model with spatial diffusion. The main results of this note are concerned with the existence and uniqueness of traveling wave solution as well as periodic wave train solution in the large wave speed asymptotic. Depending on the model parameters we more particularly study the existence and uniqueness of a traveling wave connecting two equilibria or connecting an equilibrium point and a periodic wave train. We also discuss the existence and uniqueness of such a periodic wave train. Our analysis is based on ordinary differential equation techniques by coupling the theories of invariant manifolds together with those of global attractors.

Minimal Wave Speed in a Competitive Integrodifference System without Comparison Principle

We investigate the traveling wave solutions of a competitive integrodifference system without comparison principle. In the earlier conclusions, a threshold of wave speed is defined while the existence or nonexistence of traveling wave solutions remains open when the wave speed is the threshold. By constructing generalized upper and lower solutions, we confirm the existence of traveling wave solutions when the wave speed is the threshold. Our conclusion completes the known results and shows the different decay behavior of traveling wave solutions compared with the case of large wave speeds.

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.

Stability of Traveling Wave Fronts for a Three Species Predator‐Prey Model with Nonlocal Dispersals

In this paper, we consider a predator‐prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decay exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy method and the squeezing technique with some new flavors to handle the nonlocal dispersals.

Traveling wavefronts in a reaction–diffusion model with chemotaxis and nonlocal delay effect

This paper is devoted to the study of traveling wavefronts of large wave speed for a reaction–diffusion model with chemotaxis and nonlocal delay effect by applying the perturbation method. The proof relies on an abstract formulation of the wave profile as a solution of an operator equation in a certain Banach space, coupled with the Fredholm theory and the Banach contraction mapping principle. This result is illustrated by an application to the chemotaxis–diffusion–growth model with the logistic source and a single delay effect.

Periodic traveling waves in a reaction–diffusion model with chemotaxis and nonlocal delay effect

In this paper, we study the existence of periodic traveling wave solutions with large wave speed for a reaction–diffusion model with chemotaxis and nonlocal delay effect by applying the perturbation method. The proof relies on an abstract formulation of the wave profile as a solution of an operator equation in a certain Banach space, coupled with the Lyapunov–Schmidt reduction and the implicit function theorem.

Stability of traveling wave fronts for a cooperative system with nonlocal dispersals

In this paper, we investigate a cooperative system with nonlocal dispersals. It is well known that the existence of traveling wave fronts is established by Li and Lin (Appl Math Comput 204:738–744, 2008). We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decays exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy and the squeezing technique with some new flavors to handle the nonlocal dispersals.