In this paper we review the history and current state-of-the-art in modelling of long nonlinear dispersive waves. For the sake of conciseness of this review we omit the unidirectional models and focus especially on some classical and improved BOUSSINESQ-type and SERRE–GREEN–NAGHDI equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

In this paper we describe a numerical method to solve numerically the weakly dispersive fully nonlinear SERRE–GREEN–NAGHDI (SGN) celebrated model. Namely, our scheme is based on reliable finite volume methods, proven to be very efficient for the hyperbolic part of equations. The particularity of our study is that we develop an adaptive numerical model using moving grids. Moreover, we use a special form of the SGN equations where non-hydrostatic part of pressure is found by solving a linear elliptic equation. Moreover, this form of governing equations allows to determine the natural form of boundary conditions to obtain a well-posed (numerical) problem.

We study correlations of hydrodynamic fluctuations in shear flow analytically and also by dissipative particle dynamics (DPD) simulations. The hydrodynamic equations are linearized around the macroscopic velocity field and then solved by a perturbation method in Fourier-transformed space. The autocorrelation functions (ACFs) from the analytical method are compared with results obtained from DPD simulations under the same shear-flow conditions. Up to a moderate shear rate, various ACFs from the two approaches agree with each other well. At large shear rates, discrepancies between the two methods are observed, hence revealing strong additional coupling between different fluctuating variables, which is not considered in the analytical approach. In addition, the results at low and moderate shear rates can serve as benchmarks for developing multiscale algorithms for coupling of heterogeneous solvers, such as a hybrid simulation of molecular dynamics and fluctuating hydrodynamics solver, where thermal fluctuations are indispensable.

In this paper, a new earthquake location method based on the waveform inversion is proposed. As is known to all, the waveform misfit function under the L² measure is suffering from the cycle skipping problem. This leads to a very small convergence domain of the conventional waveform based earthquake location methods. In present study, by introducing and solving two simple sub-optimization problems, we greatly expand the convergence domain of the waveform based earthquake location method. According to a large number of numerical experiments, the new method expands the range of convergence by several tens of times. This allows us to locate the earthquake accurately even from some relatively bad initial values.

We derive a closed form expression for the regularized Stokeslet in two space dimensions with periodic boundary conditions in the x-direction and a solid plane wall at y = 0. To accommodate the no-slip condition on the wall, a system of images for the regularized Stokeslets was used. The periodicity is enforced by writing all elements of the image system in terms of a Green's function whose periodic extension is known. Although the formulation is derived in the context of regularized Stokeslets, the expression for the traditional (singular) Stokeslet is easily found by taking the limit as the regularization parameter approaches zero. The new formulation is validated by comparing results of two test problems: the Taylor infinite waving sheet and the motion of a cylinder moving near a wall. As an example of an application, we use our formulation to compute the motion and flow generated by cilia using a model that does not prescribe the motion so that the beat period and synchronization of neighboring cilia are a result of the forces developed along the cilia.

This paper is devoted to the error estimate for the iterative discontinuous Galerkin (IDG) method introduced in [P. Yin, Y. Huang and H. Liu. Commun. Comput. Phys. 16: 491–515, 2014] to the nonlinear Poisson-Boltzmann equation. The total error includes both the iteration error and the discretization error of the direct DG method to linear elliptic equations. For the DDG method, the energy error is obtained by a constructive approach through an explicit global projection satisfying interface conditions dictated by the choice of numerical fluxes. The $L^2$ error of order O(h^m+1) for polynomials of degree m is further recovered. The bounding constant is also shown to be independent of the iteration times. Numerical tests are given to validate the established convergence theory.

We study the asymptotic-preserving fully discrete schemes for non-equilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry. The research is based on two-temperature models with Larsen's flux-limited diffusion operators. Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation. Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors. The harmonic average approach in spherical geometry is analyzed, and its second order accuracy is demonstrated. By formal analysis, we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptotic-preserving properties. By designing associated manufactured solutions and reference solutions, we verify the desired performance of the fully discrete schemes with numerical tests, which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate, hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.

In this paper, we describe how to construct a finite-difference shock-capturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domain Ω, possibly with moving boundary. The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls. Although the motion of the boundary could be coupled with the fluid, all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow. The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domain Ω_R ⊃ Ω. We identify inner and ghost points. The inner points are the grid points located inside Ω, while the ghost points are the grid points that are outside Ω but have at least one neighbor inside Ω. The evolution equations for inner points data are obtained from the discretization of the governing equation, while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables (density, velocities and pressure). Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains. Several numerical experiments are conducted to illustrate the validity of the method. We demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the recent second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] for pure transmission problems and extend it to settings with essential boundary conditions. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in L²(Σ), where Σ denotes the union of material interfaces. To suppress spurious resonances, we introduce a combined-field version (CFIE) of our new method.
Thorough numerical tests highlight the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

We present version 1.2.0 of DASHMM, a general library implementing hierarchical multipole methods using the asynchronous multi-tasking HPX-5 runtime system. Compared with the previous release [10], this new version: (1) enables execution in both shared and distributed memory architectures; (2) extends DASHMM's infrastructure to support advanced multipole methods [18]; and (3) provides built-in implementations of both the Yukawa [15] potential and Helmholtz [16] potential in the low frequency regime. These additions have not impacted the user interface, which remains simple and extensible.