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Discrete & Continuous Dynamical Systems - B

January 2019 , Volume 24 , Issue 1

Special issue for the commemoration to create SIAM Central States Section

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Xiaoming He, Eric Kaufmann, Steve Pankavich and Erik Van Vleck
2019, 24(1): ⅰ-ⅰ doi: 10.3934/dcdsb.201901i +[Abstract](76) +[HTML](54) +[PDF](81.02KB)
Modulus metrics on networks
Nathan Albin, Nethali Fernando and Pietro Poggi-Corradini
2019, 24(1): 1-17 doi: 10.3934/dcdsb.2018161 +[Abstract](390) +[HTML](245) +[PDF](582.33KB)

The concept of \begin{document}$p$\end{document}-modulus gives a way to measure the richness of a family of objects on a graph. In this paper, we investigate the families of connecting walks between two fixed nodes and show how to use \begin{document}$p$\end{document}-modulus to form a parametrized family of graph metrics that generalize several well-known and widely-used metrics. We also investigate a characteristic of metrics called the "antisnowflaking exponent" and present some numerical findings supporting a conjecture about the new metrics. We end with explicit computations of the new metrics on some selected graphs.

Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems
Mahboub Baccouch
2019, 24(1): 19-54 doi: 10.3934/dcdsb.2018104 +[Abstract](956) +[HTML](504) +[PDF](16473.22KB)

In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the first-and second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p+3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

A Comparison of some numerical conformal mapping methods for simply and multiply connected domains
Mohamed Badreddine, Thomas K. DeLillo and Saman Sahraei
2019, 24(1): 55-82 doi: 10.3934/dcdsb.2018100 +[Abstract](771) +[HTML](625) +[PDF](2843.59KB)

This paper compares some methods for computing conformal maps from simply and multiply connected domains bounded by circles to target domains bounded by smooth curves and curves with corners. We discuss the use of explicit preliminary maps, including the osculation method of Grassmann, to first conformally map the target domain to a more nearly circular domain. The Fourier series method due to Fornberg and its generalization to multiply connected domains are then applied to compute the maps to the nearly circular domains. The final map is represented as a composition of the Fourier/Laurent series with the inverted explicit preliminary maps. A method for systematically removing corners with power maps is also implemented and composed with the Fornberg maps. The use of explict maps has been suggested often in the past, but has rarely been carefully studied, especially for the multiply connected case. Using Fourier series to represent conformal maps from domains bounded by circles to more general domains has certain computational advantages, such as the use of fast methods. However, if the target domain has elongated sections or corners, the mapping problems can suffer from severe ill-conditioning or loss of accuracy. The purpose of this paper is to illustrate some of these practical possibilites and limitations.

Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping
Belinda A. Batten, Hesam Shoori, John R. Singler and Madhuka H. Weerasinghe
2019, 24(1): 83-107 doi: 10.3934/dcdsb.2018162 +[Abstract](438) +[HTML](259) +[PDF](1013.4KB)

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Convergence rates for semistochastic processes
James Broda, Alexander Grigo and Nikola P. Petrov
2019, 24(1): 109-125 doi: 10.3934/dcdsb.2019001 +[Abstract](105) +[HTML](76) +[PDF](499.04KB)

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

A dimension splitting and characteristic projection method for three-dimensional incompressible flow
Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen and Yiren Yang
2019, 24(1): 127-147 doi: 10.3934/dcdsb.2018111 +[Abstract](761) +[HTML](513) +[PDF](4863.68KB)

A dimension splitting and characteristic projection method is proposed for three-dimensional incompressible flow. First, the characteristics method is adopted to obtain temporal semi-discretization scheme. For the remaining Stokes equations we present a projection method to deal with the incompressibility constraint. In conclusion only independent linear elliptic equations need to be calculated at each step. Furthermore on account of splitting property of dimension splitting method, all the computations are carried out on two-dimensional manifolds, which greatly reduces the difficulty and the computational cost in the mesh generation. And a coarse-grained parallel algorithm can be also constructed, in which the two-dimensional manifold is considered as the computation unit.

A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations
Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang and Steven M. Wise
2019, 24(1): 149-182 doi: 10.3934/dcdsb.2018090 +[Abstract](837) +[HTML](486) +[PDF](4808.07KB)

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an \begin{document}$\ell^2 (0, T; H_h^3)$\end{document} stability of the numerical scheme. To overcome the difficulty associated with the convection term \begin{document}$\nabla · (\phi \mathit{\boldsymbol{u}})$\end{document}, we perform an \begin{document}$\ell^∞ (0, T; H_h^1)$\end{document} error estimate instead of the classical \begin{document}$\ell^∞ (0, T; \ell^2)$\end{document} one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.

Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces
Chao Deng and Tong Li
2019, 24(1): 183-195 doi: 10.3934/dcdsb.2018093 +[Abstract](690) +[HTML](446) +[PDF](427.83KB)

This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the \begin{document}$L^1$\end{document} and \begin{document}$L^∞$\end{document} spaces, we first prove global well-posedness of the system in \begin{document}$L^1× L^∞$\end{document} which partially answers the question posted by Kozono et al in [19]. For the case \begin{document}$μ_0>0$\end{document}, we make full use of the linear parts of the system to get the improved long time decay property. Moreover, by using the new formulation involving all linear parts, introducing the logarithmic-weight in time to modify the other endpoint space \begin{document}$L^∞× L^∞$\end{document}, and carefully decomposing time into several pieces, we are able to establish the global well-posedness and large time behavior of the system in \begin{document}$L^∞_{ln}× L^∞$\end{document}.

Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Aslihan Demirkaya and Milena Stanislavova
2019, 24(1): 197-209 doi: 10.3934/dcdsb.2018097 +[Abstract](719) +[HTML](487) +[PDF](2578.09KB)

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: \begin{document}$u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$\end{document} when \begin{document}$p = 3$\end{document} and \begin{document}$p = 5$\end{document}. For the standing wave solutions \begin{document}$u(x, t) = e^{iω t}\varphi_{ω}(x)$\end{document} we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value \begin{document}$ω^*$\end{document} which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions \begin{document}$u(x, t) = φ_c(x+ct)$\end{document}. We present numerical illustration of the theoretically predicted threshold value of the speed \begin{document}$c$\end{document} which separates the stable and unstable waves.

Global regularity results for the climate model with fractional dissipation
Boqing Dong, Wenjuan Wang, Jiahong Wu and Hui Zhang
2019, 24(1): 211-229 doi: 10.3934/dcdsb.2018102 +[Abstract](621) +[HTML](419) +[PDF](465.67KB)

This paper studies the global well-posedness problem on a tropical climate model with fractional dissipation. This system allows us to simultaneously examine a family of equations characterized by the fractional dissipative terms \begin{document}$ (-Δ)^{\mathit{\alpha }}u$\end{document} in the equation of the barotropic mode \begin{document}$ u$\end{document} and \begin{document}$ (-Δ)^β v$\end{document} in the equation of the first baroclinic mode \begin{document}$ v$\end{document}. We establish the global existence and regularity of the solutions when the total fractional power is 2, namely \begin{document}$ {\mathit{\alpha }}+ β = 2$\end{document}.

A comparative study on nonlocal diffusion operators related to the fractional Laplacian
Siwei Duo, Hong Wang and Yanzhi Zhang
2019, 24(1): 231-256 doi: 10.3934/dcdsb.2018110 +[Abstract](864) +[HTML](767) +[PDF](1850.41KB)

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as \begin{document}$ \alpha \to2 $\end{document}. The eigenvalues and eigenfunctions of these four operators are different, and the \begin{document}$ k $\end{document}-th (for \begin{document}$ k \in {\mathbb N} $\end{document}) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \begin{document}$ \alpha \in (0, 2) $\end{document}, the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \begin{document}$ \delta $\end{document} is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of \begin{document}$ {\mathcal O}(\delta ^{-\alpha }) $\end{document}. In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \begin{document}$ \alpha \to2 $\end{document}, it generally provides inconsistent result from that of the fractional Laplacian if \begin{document}$ \alpha \ll 2 $\end{document}. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model
Mahadevan Ganesh, Brandon C. Reyes and Avi Purkayastha
2019, 24(1): 257-272 doi: 10.3934/dcdsb.2018107 +[Abstract](653) +[HTML](544) +[PDF](739.09KB)

We consider a moving shutter and non-deterministic generalization of the diffraction in time model introduced by Moshinsky several decades ago to study a class of quantum transients. We first develop a moving-mesh finite element method (FEM) to simulate the determisitic version of the model. We then apply the FEM and multilevel Monte Carlo (MLMC) algorithm to the stochastic moving-domain model for simulation of approximate statistical moments of the density profile of the stochastic transients.

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection
Iuliana Oprea and Gerhard Dangelmayr
2019, 24(1): 273-296 doi: 10.3934/dcdsb.2018095 +[Abstract](672) +[HTML](481) +[PDF](8936.92KB)

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with \begin{document}$O(2) × O(2)$\end{document} symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.

Predicting and estimating probability density functions of chaotic systems
Noah H. Rhee, PaweŁ Góra and Majid Bani-Yaghoub
2019, 24(1): 297-319 doi: 10.3934/dcdsb.2017144 +[Abstract](1407) +[HTML](783) +[PDF](482.85KB)

In the present work, for the first time, we employ Ulam's method to estimate and to predict the existence of the probability density functions of single species populations with chaotic dynamics. In particular, given a chaotic map, we show that Ulam's method generates a sequence of density functions in L1-space that may converge weakly to a function in L1-space. In such a case we show that the limiting function generates an absolutely continuous (w.r.t. the Lebesgue measure) invariant measure (w.r.t. the given chaotic map) and therefore the limiting function is the probability density function of the chaotic map. This fact can be used to determine the existence and estimate the probability density functions of chaotic biological systems.

Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics
Naveen K. Vaidya, Xianping Li and Feng-Bin Wang
2019, 24(1): 321-349 doi: 10.3934/dcdsb.2018099 +[Abstract](767) +[HTML](556) +[PDF](997.36KB)

In recent years, the growing spatial spread of dengue, a mosquito-borne disease, has been a major international public health concern. In this paper, we propose a mathematical model to describe an impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. We first consider homogeneous temperature profiles across space and study sensitivity of the basic reproduction number to the environmental temperature. We then introduce spatially heterogeneous temperature into the model and establish some important properties of dengue dynamics. In particular, we formulate two indices, mosquito reproduction number and infection invasion threshold, which completely determine the global threshold dynamics of the model. We also perform numerical simulations to explore the impact of spatially heterogeneous temperature on the disease dynamics. Our analytical and numerical results reveal that spatial heterogeneity of temperature can have significant impact on expansion of dengue epidemics. Our results, including threshold indices, may provide useful information for effective deployment of spatially targeted interventions.

A locking free Reissner-Mindlin element with weak Galerkin rotations
Ruishu Wang, Lin Mu and Xiu Ye
2019, 24(1): 351-361 doi: 10.3934/dcdsb.2018086 +[Abstract](694) +[HTML](453) +[PDF](379.2KB)

A locking free finite element method is developed for the Reissner-Mindlin equations in their primary form. In this method, the transverse displacement is approximated by continuous piecewise polynomials of degree \begin{document} $k+1$ \end{document} and the rotation is approximated by weak Galerkin elements of degree \begin{document} $k$ \end{document} for \begin{document} $k≥1$ \end{document}. A uniform convergence in thickness of the plate is established for this finite element approximation. The numerical examples demonstrate locking free of the method.

Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains
Xinguang Yang, Baowei Feng, Thales Maier de Souza and Taige Wang
2019, 24(1): 363-386 doi: 10.3934/dcdsb.2018084 +[Abstract](714) +[HTML](467) +[PDF](487.19KB)

This article focuses on the optimal regularity and long-time dynamics of solutions of a Navier-Stoke-Voigt equation with non-autonomous body forces in non-smooth domains. Optimal regularity is considered, since the regularity \begin{document} $H_0^1\cap H^2$ \end{document} cannot be achieved. Given the initial data in certain spaces, it can be shown that the problem generates a well-defined evolutionary process. Then we prove the existence of a uniform attractor consisting of complete trajectories.

Two-grid finite element method for the stabilization of mixed Stokes-Darcy model
Jiaping Yu, Haibiao Zheng, Feng Shi and Ren Zhao
2019, 24(1): 387-402 doi: 10.3934/dcdsb.2018109 +[Abstract](1092) +[HTML](589) +[PDF](4460.98KB)

A two-grid discretization for the stabilized finite element method for mixed Stokes-Darcy problem is proposed and analyzed. The lowest equal-order velocity-pressure pairs are used due to their simplicity and attractive computational properties, such as much simpler data structures and less computer memory for meshes and algebraic system, easier interpolations, and convenient usages of many existing preconditioners and fast solvers in simulations, which make these pairs a much popular choice in engineering practice; see e.g., [4,27]. The decoupling methods are adopted for solving coupled systems based on the significant features that decoupling methods can allow us to solve the submodel problems independently by using most appropriate numerical techniques and preconditioners, and also to reduce substantial coding tasks. The main idea in this paper is that, on the coarse grid, we solve a stabilized finite element scheme for coupled Stokes-Darcy problem; then on the fine grid, we apply the coarse grid approximation to the interface conditions, and solve two independent subproblems: one is the stabilized finite element method for Stokes subproblem, and another one is the Darcy subproblem. Optimal error estimates are derived, and several numerical experiments are carried out to demonstrate the accuracy and efficiency of the two-grid stabilized finite element algorithm.

Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes
Qilong Zhai and Ran Zhang
2019, 24(1): 403-413 doi: 10.3934/dcdsb.2018091 +[Abstract](713) +[HTML](497) +[PDF](371.2KB)

In this paper, we investigate the weak Galerkin method for the Laplacian eigenvalue problem. We use the weak Galerkin method to obtain lower bounds of Laplacian eigenvalues, and apply a postprocessing technique to get upper bounds. Thus, we can verify the accurate intervals which the exact eigenvalues lie in. This postprocessing technique is efficient and does not need to solve any auxiliary problem. Both theoretical analysis and numerical experiments are presented in this paper.

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