The present article is the third part of a series of papers devoted to the shallow water wave modelling. In this part we investigate the derivation of some long wave models on a deformed sphere. We propose first a suitable for our purposes formulation of the full EULER equations on a sphere. Then, by applying the depth-averaging procedure we derive first a new fully nonlinear weakly dispersive base model. After this step we show how to obtain some weakly nonlinear models on the sphere in the so-called BOUSSINESQ regime. We have to say that the proposed base model contains an additional velocity variable which has to be specified by a closure relation. Physically, it represents a dispersive correction to the velocity vector. So, the main outcome of our article should be rather considered as a whole family of long wave models.

In the present manuscript we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniform elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealized situations with even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the BULGARIAN 2007 and CHILEAN 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements.

In this work, we develop an hp-adaptivity strategy for the minimum action method (MAM) using a posteriori error estimate. MAM plays an important role in minimizing the Freidlin-Wentzell action functional, which is the central object of the Freidlin-Wentzell theory of large deviations for noise-induced transitions in stochastic dynamical systems. Because of the demanding computation cost, especially in spatially extended systems, numerical efficiency is a critical issue for MAM. Difficulties come from both temporal and spatial discretizations. One severe hurdle for the application of MAM to large scale systems is the global reparametrization in time direction, which is needed in most versions of MAM to achieve accuracy. We recently introduced a new version of MAM in [22], called tMAM, where we used some simple heuristic criteria to demonstrate that tMAM can be effectively coupled with $h$-adaptivity, i.e., the global reparametrization can be removed. The target of this paper is to integrate $hp$-adaptivity into tMAM using a posteriori error estimation techniques, which provides a general adaptive MAM more suitable for parallel computing. More specifically, we use the zero-Hamiltonian constraint to define an indicator to measure the error induced by linear time scaling, and the derivative recovery technique to construct an error indicator and a regularity indicator for the transition paths approximated by finite elements. Strategies for $hp$-adaptivity have been developed. Numerical results are presented.

Cortical spreading depression (CSD) involves depolarization of neurons and astrocytes due principally to a large increase in extracellular potassium. CSD is also accompanied by large increases in extracellular ATP and is blocked by glutamate N-methyl-D-aspartate (NMDA) receptor antagonists. In a previous study, we used a neuronal model to investigate the instigation of CSD and propagation of CSD wave by increasing extracellular potassium. In this paper, we extend that model by constructing a neuron-astrocyte network and incorporating the effects of ATP via the coupled biochemical pathways involving glutamate and NMDA currents. We show that both the electrical current stimuli and the local elevation in the extracellular ATP or glutamate concentration can lead to CSD in the coupled neuron-astrocyte model, while ATP or glutamate increase fails to induce CSD when the glutamate NMDA channels are blocked. These results can explain both the potassium theory postulated by Grafstein and glutamate theory by Van Harreveld. Our model showed extracellular potassium plays the key role in CSD instigation and propagation, and glutamate is a key compound mediating CSD by activating NMDA channels through neuron-astrocyte interactive.

We present a general framework of the theory of spontaneous symmetry breaking in non-relativistic systems. We discuss a spontaneous symmetry breaking in a system with general global symmetry given by a Lie group G. The Nambu-Goldstone boson and Higgs boson are represented explicitly by local fields by means of the basis of the Lie algebra of G. An application to superconductivity is discussed. We evaluate the Green's functions of the Nambu-Goldstone and Higgs bosons in superconductors. We show that the Nambu-Goldstone and Higgs modes exhibit interesting behaviors in multi-component superconductors.

Mortar methods are widely used techniques for discretizations of partial differential equations and preconditioners for the algebraic systems resulting from the discretizations. For problems with high contrast and multiple scales, the standard mortar spaces are not robust, and some enrichments are necessary in order to obtain an efficient and robust mortar space. In this paper, we consider a class of flow problems in high contrast heterogeneous media, and develop a systematic approach to obtain an enriched multiscale mortar space. Our approach is based on the constructions of local multiscale basis functions. The multiscale basis functions are constructed from local problems by following the framework of the Generalized Multiscale Finite Element Method (GMsFEM). In particular, we first create a local snapshot space. Then we select the dominated modes within the snapshot space using an appropriate Proper Orthogonal Decomposition (POD) technique. These multiscale basis functions show better accuracy than polynomial basis for multiscale problems. Using the proposed multiscale mortar space, we will construct a multiscale finite element method to solve the flow problem on a coarse grid and a preconditioning technique for the fine scale discretization of the flow problem. In particular, we develop a multiscale mortar mixed finite element method using the mortar space. In addition, we will design a two-level additive preconditioner and a two-level hybrid preconditioner based on the proposed mortar space for the iterative method applied to the fine scale discretization of the flow problem. We present several numerical examples to demonstrate the efficiency and robustness of our proposed mortar space with respect to both the coarse multiscale solver and the preconditioners.

A relativistic generalization of the inviscid Burgers equation was introduced by LeFloch and co-authors and was recently investigated numerically on a Schwarzschild background. We extend this analysis to a Friedmann-Lemaître-Robertson-Walker (FLRW) background, which is more challenging due to the existence of time-dependent, spatially homogeneous solutions. We present a derivation of the model of interest and we study its basic properties, including the class of spatially homogeneous solutions. Then, we design a second-order accurate scheme based on the finite volume methodology, which provides us with a tool for investigating the properties of solutions. Computational experiments demonstrate the efficiency of the proposed scheme for numerically capturing weak solutions.

The p$K_a$ values are important quantities characterizing the ability of protein active sites to give up protons. p$K_a$ can be measured using NMR by tracing chemical-shifts of some special atoms, which is however expensive and time-consuming. Alternatively, p$K_a$ can be calculated numerically by electrostatic free energy changes subject to the protonation and deprotonation of titration sites. To this end, the Poisson-Boltzmann (PB) model is an effective approach for the electrostatics. However, numerically solving PB equation is challenging due to the jump conditions across the dielectric interfaces, irregular geometry of the molecular surface, and charge singularities. Our recently developed matched interface and boundary (MIB) method treats these challenges rigorously, resulting in a solid second order MIBPB solver. Since the MIBPB solver uses Green's function based regularization of charge singularities by decomposing the solution into a singular component and a regularized component, it is particularly efficient in treating the accuracy-sensitive, numerous, and complicated charge distributions from the p$K_a$ calculation. Our numerical results demonstrate that accurate free energies and p$K_a$ values are achieved at coarse grid rapidly. In addition, the resulting software, which pipelines the entire p$K_a$ calculation procedure, is available to all potential users from the greater bioscience community.

Inspired by [Q. Y. Hu, S. Shu and J. X. Wang, Math. Comput., 79 (272) (2010): 2059-2078], we firstly present two nonoverlapping domain decomposition (DD) preconditioners $B^a_h$ and $B^{sm}_h$ about the preserving-symmetry finite volume element (SFVE) scheme for solving two-dimensional three-temperature radiation diffusion equations with strongly discontinuous coefficients. It's worth mentioning that both $B^a_h$ and $B^{sm}_h$ involve a SFVE sub-system with respect to a simple coarse space and SFVE sub-systems which are self-similar to the original SFVE system but embarrassingly parallel. Next, the nearly optimal estimation $\mathcal{O}$((1+log$\frac{d}{h}$)³) on condition numbers is proved for the resulting preconditioned systems, where d and h respectively denote the maximum diameters in coarse and fine grids. Moreover, we present algebraic and parallel implementations of  $B^a_h$ and $B^{sm}_h$, develop parallel PCG solvers, and provide the numerical results validating the aforementioned theoretical estimations and stating the good algorithmic and parallel scalabilities.

Focused transducers are frequently used in photoacoustic endoscopy (PAE) to ensure high acoustic resolution beyond the depth of optical diffraction. Although most transducers in PAE are ceramic and focus-fixed, we here propose the polyvinylidine fluoride (PVDF) based annular array to reduce the fabrication cost, and to allow dynamic focusing for increased depth of field. In this work, a method of direct lithography on the aluminum coated PVDF films was used for the annular array fabrication. The PVDF annular array was built with a 110 µm thick aluminum coated PVDF film, which contained three independent rings of equal width within a 5.5 mm aperture size. Phantom experiments were conducted for field test, and a simple linear model was used to compare the acquired results in the acoustic field test. The device was used for demonstration in PAE application. Potential applications of the proposed method include the PAE imaging of several large human organs and lumens such as the gastrointestinal tract and the cervical canal.

In this paper we present a second order accurate (in time) energy stable numerical scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation in space. Instead of the standard second order Crank-Nicolson methodology, we apply the implicit backward differentiation formula (BDF) concept to derive second order temporal accuracy, but modified so that the concave diffusion term is treated explicitly. This explicit treatment for the concave part of the chemical potential ensures the unique solvability of the scheme without sacrificing energy stability. An additional term $A$τ∆($u^{k+1}$−$u^k$) is added, which represents a second order Douglas-Dupont-type regularization, and a careful calculation shows that energy stability is guaranteed, provided the mild condition $A$≥$\frac{1}{16}$ is enforced. In turn, a uniform in time $H^1$ bound of the numerical solution becomes available. As a result, we are able to establish an $ℓ^∞$(0,$T$;$L^2$) convergence analysis for the proposed fully discrete scheme, with full $\mathcal{O}$ (τ²+$h^2$) accuracy. This convergence turns out to be unconditional; no scaling law is needed between the time step size $τ$ and the spatial grid size $h$. A few numerical experiments are presented to conclude the article.

Recently, Garcke et al. [H. Garcke, M. Hinze, C. Kahle, Appl. Numer. Math. 99 (2016), 151–171)] developed a consistent discretization scheme for a thermodynamically consistent diffuse interface model for incompressible two-phase flows with different densities [H. Abels, H. Garcke, G. Grün, Math. Models Methods Appl. Sci. 22(3) (2012)]. At the heart of this method lies the solution of large and sparse linear systems that arise in a semismooth Newton method. 

In this work we propose the use of preconditioned Krylov subspace solvers using effective Schur complement approximations. Numerical results illustrate the efficiency of our approach. In particular, our preconditioner is shown to be robust with respect to parameter changes.