In this paper, we propose a stable, convergent finite difference scheme to solve numerically a nonlocal phase field system which may model a variety of nonisothermal phase separations in pure materials which can assume two different phases, say solid and liquid, with properties varying in space. The scheme inherits the characteristic property of conservation of internal energy. We also prove that the scheme is uniquely solvable and the numerical solution will approach the true solution in the L^∞-norm.

This work is a continuation of Kou and Sun [36] where we present an efficient improvement on the IMplicit Pressure Explicit Saturation (IMPES) method for two-phase immiscible fluid flow in porous media with different capillarity pressures. In the previous work, we present an implicit treatment of capillary pressure appearing in the pressure equation. A linear approximation of capillary function is used to couple the implicit saturation equation into the pressure equation that is solved implicitly. In this paper, we present an iterative version of this method. It is well-known that the fully implicit scheme has unconditional stability. The new method can be used for solving the coupled system of nonlinear equations arisen after the fully implicit scheme. We follow the idea of the previous work, and use the linear approximation of capillary function at the current iteration. This is different from iterative IMPES that computes capillary pressure by the saturations at the previous iteration. From this approximation, we couple the saturation equation into the pressure equation, and establish the coupling relation between the pressure and saturation. We employ the relaxation technique to control the convergence of the new method, and we give a choice of relaxation factor. The convergence theorem of our method is established under the natural conditions. Numerical examples are provided to demonstrate the performance of our approach, and the results show that our method is efficient and stable.

This paper discusses the multiscale method for the time-dependent Maxwell's equations with memory effects in composite materials. The main difficulty is that one cannot use the usual multiscale asymptotic method (cf. [25, 4]) to solve this problem, due to the complication of the memory terms. The key steps addressed in this paper are to transfer the original integro-differential equations to the stationary Maxwell's equations by using the Laplace transform, to employ the multiscale asymptotic method to solve the stationary Maxwell's equations, and then to obtain the computational solution of the original problem by employing a quadrature formula for computing the inverse Laplace transform. Numerical simulations are then carried out to validate the multiscale numerical algorithm in the present paper.

We discuss the complete consensus problem of the discrete-time dynamical networks with time-varying couplings, and provide a simple analytic proof for the emergence of asymptotic complete consensus. Our approach is based on the “energy estimate argument” and connectivity of the communication topology. As direct application of our main results, we obtain asymptotic complete consensus for the discrete-time Kuramoto model with local communication topology.

For the initial-boundary value problem to the 2 × 2 damped p-system with nonlinear source, \begin{equation*} \begin{cases} &v_t-u_x=0, \\ &u_t+p(v)_x=-α u-β |u|^{q-1}u,q\;\geq\;2 \\ &(v, u)|_t=0(v_0, u_0)(x) → (v_+, u_+)\;\;as x → +∞,\\ &u_{x=0}=0, u_+\neq0, \end{cases} \quad (x,t)∈\mathbb{R}_+×\mathbb{R}_+, \end{equation*} when β › 0, or β ‹ 0 but |β| ‹ \frac{α}{|u_+|^{q-1}}, the solution (v, u)(x, t) is proved to globally exist and converge to the solution of the corresponding porous media equations \begin{equation*} \begin{cases} &\bar{v}_t-\bar{u}_x=0, \\ &p_(\bar{v})_x=-α\bar{u}, \\ &\bar{v}|_t=0\bar{v}0(x) → v_+\;\;as x →+∞,\\ &\bar{v}|_x=0, \end{cases} \quad (x,t)\in\mathbb{R}_+\times\mathbb{R}_+, \end{equation*} with a specially selected initial data \bar{v}0(x). The optimal convergence rates \|\partial^k_x(v-\bar{v},u-\bar{u})(t)\|_L^2=0(1)(t^{-\frac{2k+3}{4}}, t^{-\frac{2k+5}{4}}), k = 0, 1, are also obtained, as the initial perturbation is in L^1(\mathbb{R}_+)\cap H^3(\mathbb{R}_+). If the initial perturbation is in the weighted space L^{1,\gamma}(\mathbb{R}_+)\cap H^3(\mathbb{R}_+) with the best choice of \gamma=\frac{1} {4} , some new and much better decay rates are further obtained: \|\partial^k_x(v-\bar{v})(t)\|_L^2=0(1+t)^{-\frac{2k+3}{4}-\frac{\gamma}{2}}, k = 0, 1. The proof is based on the technical weighted energy method combining with the Green function method. However, when β ‹ 0 and |β| ›  \frac{α}{|u_+|^{q-1}}, then the solution will blow up at a finite time. Finally, numerical simulations are carried out to confirm the theoretical results by using the central-upwind scheme. In particular, the interest phenomenon of coexistence of the global solution v(x, t) and the blow-up solution u(x, t) is observed and numerically demonstrated.

Resource-consumer models have been applied to explain population cycles of small mammals such as brown lemmings in Alaska. All these models only consider food quantity for small mammals. However, food quality can potentially be a key factor driving the population cycle. To capture both food quantity and quality in the resource-consumer model, we apply the newly emerged method “ecological stoichiometry”, which deals with the balance of fundamental elements in living organisms. A group of stoichiometric models are discussed in this paper for brown lemmings in Alaska, where food quality is measured by phosphorus and food quantity is measured by carbon. Within the framework of our models, we define an index to compare the relative importance of food quality and food quantity. Simulations of this index show that brown lemming cycles in Alaska are mainly controlled by food quantity. Bifurcation diagrams illustrate that the cycle period is an increasing function of the nutrient availability but a decreasing function of the nutrient requirement of lemmings. A striking result arises: high nutrient availability and small nutrient requirement of lemmings drive the low points of the population cycle to be extremely small, leading to high probability of lemmings' extinction. However, high nutrient availability and small nutrient requirement of lemmings should both benefit lemmings. This paradox needs further examination in theoretical and empirical studies. In addition, we perform sensitivity analysis of periodicity with respect to all parameters.

A smooth block LU factorization, coupled with Newton's method, is used to compute multiple nonlinear eigenvalues of smooth rank-deficient matrix functions A($\lambda$). We provide conditions for such factorizations to exist and show that the algorithm for the computation of multiple nonlinear eigenvalues converges quadratically, and is more efficient than one using QR factorizations. A possible approach for cubic convergence is also discussed. Several numerical examples are given for general and random nonlinear matrix functions A($\lambda$).