We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$

We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

A partition calculation method (PCM) for calculating the diffraction efficiency of multilayer Fresnel zone plate with high aspect ratio is proposed. In contrast to the traditional theory, PCM designs and evaluates Fresnel zone plate (FZP) considering material pairs, all zone widths, thicknesses and X-ray energy more completely. The results obtained through PCM are validated by comparing them with the complex amplitude superposition theory and coupled wave theory numerical results. The PCM satisfies the requirements of the theoretical investigation of FZP with small outermost zone width (drN) and large thickness (t). Combining proper numerical analysis with the experimental conditions will present a great potential to break through the imaging performance of X-ray microscopy.

Hermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.

This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legendre 2-point formula for numerical integration, chemical equilibrium application, kinematic application, neuropsychology application, combustion application and interval arithmetic benchmark) in order to evaluate the performance of the new approach. Empirical results reveal that the proposed method is characterized by fast convergence and is able to deal with high-dimensional equations systems.

A numerical investigation of an electroosmotic flow through a microchannel is presented. Lattice Poisson-Boltzmann method was utilized to determine the effective geometrical and electrokinetic parameters in a microfluidic system. The non-Newtonian fluid model is assumed to be viscoplastic which is suitable for modeling biologic structures. These types of fluids are shown to have a yield stress which affects the velocity profile significantly. Unlike Casson fluid constitutive properties, electrokinetic parameters are shown not to be effective on the yielded region in the microchannel. The influence of flow and viscokinetic parameters on yield height, plug-flow velocity and mass flow rate was studied and discussed.

Metallic phase-change materials (PCMs) attract much attention due to their high thermal conductivity in thermal energy storage. Our previous work reported a kind of Cu@Cr@Ni bilayer capsules, which could endure at least 1000 thermal cycles between 1323 and 1423 K without leakage, and might be a potential high-temperature metallic PCM. This study numerically investigates the thermal energy charging performance of Cu@Cr@Ni capsules for recovering high-temperature waste heat at both constant and periodically fluctuant heat transfer fluid temperatures. It was revealed that only a short and slight sloped melting platform existed in the curve of outlet temperature due to the ultrahigh thermal conductivity of copper; with higher inlet velocities, the outlet and mean temperatures of such PCM increased and meanwhile the energy transfer efficiency decreased; the outlet and mean temperatures of the PCM and the liquid fraction in it were rather insensitive to the period of the inlet temperature fluctuation; and the amplitude of inlet temperature fluctuation, ±50 K, was sharply reduced to 5 K due to the thermal damping of the PCM.

Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

In the present study, a 3-dimensional model was developed to investigate fluid flow in MHD micro-pumps. Initially, 3D governing equations were derived and numerically solved using the finite volume method/SIMPLE algorithm. The case study was a (MHD) micropump built in the year 2000 (channel length: 20mm, channel width: 800μm, channel height: 380μm and electrode length: 4mm). The applied magnetic flux density was 13mT and the electric current was different for various solutions (10 ~ 140mA). The numerical results were verified by experimental and analytical data for several solutions. In addition effects of magnetic field strength, electric current, geometrical parameters of the MHD micropump, electrode length and electrode location on its performance have been investigated. Finally the results has been considered and discussed.

The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.

The importance of software continues to grow for all areas of scientific research, no less for powder diffraction. Knowing how to program a computer is a basic and useful skill for scientists. This paper explains the three approaches for programming languages and why scripting languages are preferred for non-expert programmers. The Python-scripting language is extremely efficient for science and its use by scientists is growing. Python is also one of the easiest languages to learn. The language is introduced, as well as a few of the many add-on packages available that extend its capabilities, for example, for numerical computations, scientific graphics, and graphical user interface programming. Resources for learning Python are also provided.

The Courant-Friedrichs-Lewy condition (The CFL condition) is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations. We give a remark on the CFL condition from a view point of stability, and we give some numerical experiments which show instability of numerical solutions even under the CFL condition. We give a mathematical model for rounding errors in order to explain the instability.

A new class of history-dependent quasivariational inequalities was recently studied in[M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising incontact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491].Existence, uniqueness and regularity results were proved and used in the study of severalmathematical models which describe the contact between a deformable body and an obstacle.The aim of this paper is to provide numerical analysis of the quasivariationalinequalities introduced in the aforementioned paper. To this end we introduce temporallysemi-discrete and fully discrete schemes for the numerical approximation of theinequalities, show their unique solvability, and derive error estimates. We then applythese results to a quasistatic frictional contact problem in which the material’s behavioris modeled with a viscoelastic constitutive law, the contact is bilateral, and friction isdescribed with a slip-rate version of Coulomb’s law. We discuss implementation of thecorresponding fully-discrete scheme and present numerical simulation results on atwo-dimensional example.

The Coupled Cluster (CC) method is a widely used and highly successful high precisionmethod for the solution of the stationary electronic Schrödingerequation, with its practical convergence properties being similar to that of acorresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method beenanalyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in[Schneider, 2009]. Recently, we globalized the CC formulation to the full continuousspace, giving a root equation for an infinite dimensional, nonlinear Coupled Clusteroperator that is equivalent the full electronic Schrödinger equation [Rohwedder, 2011]. Inthis paper, we combine both approaches to prove existence and uniqueness results,quasi-optimality estimates and energy estimates for the CC method with respect to thesolution of the full, original Schrödinger equation. The main property used is a localstrong monotonicity result for the Coupled Cluster function, and we give twocharacterizations for situations in which this property holds.

This paper reports the results of a series of full-scale drilled shaft load tests subjected to combined axial and lateral loading and lateral loading only. The tested shafts, 1.4m in diameter, were embedded 37m in sandy silt. All tested shafts were installed using reverse circulation method. The test results indicated, given the same lateral loading, 63% of pile head displacement resulted from combined load corresponded with the case of lateral loading only. The test results were compared to the numerical results of the software LPILE as well as the analytical solutions proposed by the senior author and his co-workers. The analytical results of the pile bending moments along shaft showed better results than that of LPILE.

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precisionmethod for the solution of the main equation of electronic structure calculation, thestationary electronic Schrödinger equation. Traditionally, theequations of CC are formulated as a nonlinear approximation of a Galerkin solution of theelectronic Schrödinger equation, i.e. within a given discrete subspace.Unfortunately, this concept prohibits the direct application of concepts of nonlinearnumerical analysis to obtain e.g. existence and uniqueness results orestimates on the convergence of discrete solutions to the full solution. Here, thisshortcoming is approached by showing that based on the choice of anN-dimensional reference subspace R of H1(ℝ3 ×{± 1/2}), the original, continuous electronic Schrödingerequation can be reformulated equivalently as a root equation for an infinite-dimensionalnonlinear Coupled Cluster operator. The canonical projected CC equations may then beunderstood as discretizations of this operator. As the main step, continuity properties ofthe cluster operator S and its adjoint S† asmappings on the antisymmetric energy space H1 are established.

Heat transfer during agitation of Bingham viscoplastic fluid is studied in this paper. The fluid is agitated with an anchor impeller and the heating is made by a jacketed wall of the stirred vessel. Transfers in the agitated vessel, translating hydrodynamic and thermal phenomena, are numerically predicted by means of Computational Fluid Dynamics (CFD) in transient regime. The purpose of this numerical study is to identify the rigid zones and to optimize mixing and heating performances. The Navier-Stokes and energy equations are discretized using finite volume method, and a two-dimensional analysis of the hydrodynamic and transient thermal behaviours generated in the agitated vessel are performed. Fluid rheology is modeled by the Bingham approximation and Papanastasiou’s regularization model. Results show the presence of recirculation zones and permit to explain the unpredicted Nusselt number increasing when Oldroyd number increases. This study shows also the importance of the anchor position on the size and the shape of the rigid zones and on the heating performances.