Implicit Approximation Research Articles

The Numerical Technique for the Landslide Tsunami Simulations Based on Navier–Stokes Equations

The paper presents an integral technique simulating all phases of a landslide-driven tsunami. The technique is based on the numerical solution of the system of Navier–Stokes equations for multiphase flows. The numerical algorithm uses a fully implicit approximation method, in which the equations of continuity and momentum conservation are coupled through implicit summands of pressure gradient and mass flow. The method we propose removes severe restrictions on the time step and allows simulation of tsunami propagation to arbitrarily large distances. The landslide origin is simulated as an individual phase being a Newtonian fluid with its own density and viscosity and separated from the water and air phases by an interface. The basic formulas of equation discretization and expressions for coefficients are presented, and the main steps of the computation procedure are described in the paper. To enable simulations of tsunami propagation across wide water areas, we propose a parallel algorithm of the technique implementation, which employs an algebraic multigrid method. The implementation of the multigrid method is based on the global level and cascade collection algorithms that impose no limitations on the paralleling scale and make this technique applicable to petascale systems. We demonstrate the possibility of simulating all phases of a landslide-driven tsunami, including its generation, propagation and uprush. The technique has been verified against the problems supported by experimental data. The paper describes the mechanism of incorporating bathymetric data to simulate tsunamis in real water areas of the world ocean. Results of comparison with the nonlinear dispersion theory, which has demonstrated good agreement, are presented for the case of a historical tsunami of volcanic origin on the Montserrat Island in the Caribbean Sea.

Direct numerical simulation of the laminar–turbulent transition at hypersonic flow speeds on a supercomputer

A method for direct numerical simulation of three-dimensional unsteady disturbances leading to a laminar–turbulent transition at hypersonic flow speeds is proposed. The simulation relies on solving the full three-dimensional unsteady Navier–Stokes equations. The computational technique is intended for multiprocessor supercomputers and is based on a fully implicit monotone approximation scheme and the Newton–Raphson method for solving systems of nonlinear difference equations. This approach is used to study the development of three-dimensional unstable disturbances in a flat-plate and compression-corner boundary layers in early laminar–turbulent transition stages at the free-stream Mach number M = 5.37. The three-dimensional disturbance field is visualized in order to reveal and discuss features of the instability development at the linear and nonlinear stages. The distribution of the skin friction coefficient is used to detect laminar and transient flow regimes and determine the onset of the laminar–turbulent transition.

Comparison of difference scheme with customizable dissipative properties and WENO scheme in the case of one-dimensional gas and gas-particle dynamics problems

Subject of Research.The paper presents results of comparative testing of highly stable difference scheme with customizable dissipative properties and WENO5 scheme in the case of one-dimensional gas dynamics and mechanics of gas-suspensions problems. Method. The second order difference scheme is constructed by physical processes splitting into two phases. The first one uses the central differences with adaptive reconstruction of artificial viscosity and semi implicit approximation of the sources. TVD type reconstructions are used in the second phase. Main Results. For the representative collection of test problems, the proposed scheme has confirmed the efficiency and good quality numerical solutions at the level of the WENO5 scheme with substantially lower machine time cost. The wave structure in the decay of initial discontinuity in the non-equilibrium gas suspension and the convergence to exact self-similar solution are considered. Practical Relevance. A detailed resolution of structural peculiarities of gas flows and gas-particle mixtures allows reducing the amount of experimental development of new technologies and technical devices of coating, leakage control when using a trial particulate matter with particles of micro - and nano-sizes and other areas.

Modified implicit fractional difference scheme for 2D modified anomalous fractional sub-diffusion equation

In this paper, we solve two-dimensional modified anomalous fractional sub-diffusion equation using modified implicit finite difference approximation. The stability and convergence of the proposed scheme are analyzed by the Fourier series method. We show that the scheme is unconditionally stable and the approximate solution converges to the exact solution. A numerical example is given to show the application and feasibility of the proposed scheme.

Mesh-based and meshless design and approximation of scalar functions

In engineering, geographical applications, bio-informatics, and scientific visualisation, a variety of phenomena is described by data modelled as the values of a scalar function defined on a surface or a volume, and critical points (i.e., maxima, minima, saddles) usually represent a relevant information about the input data or an underlying phenomenon. Furthermore, the distribution of the critical points is crucial for geometry processing and shape analysis; e.g., for controlling the number of patches in quadrilateral remeshing and the number of nodes of Reeb graphs and Morse–Smale complexes. In this context, we address the design of a smooth function, whose maxima, minima, and saddles are selected by the user or imported from a template (e.g., Laplacian eigenfunctions, diffusion maps). In this way, we support the selection of the saddles of the resulting function and not only its extrema, which is one of the main limitations of previous work. Then, we discuss the meshless approximation of an input scalar function by preserving its persistent critical points and its local behaviour, as encoded by the spatial distribution and shape of the level-sets. Both problems are addressed by computing an implicit approximation with radial basis functions, which is independent of the discretisation of differential operators and of assumptions on the sampling of the input domain. This approximation allows us to introduce a meshless iso-contouring and classification of the critical points, which are characterised in terms of the differential properties of the meshless approximation and of the geometry of the input surface, as encoded by its first and second fundamental form. Furthermore, the computation is performed at an arbitrary resolution by locally refining the input surface and by applying differential calculus to the meshless approximation. As main applications, we consider the approximation and analysis of scalar functions on both 3D shapes and volumes in graphics, Geographic Information Systems, medicine, and bio-informatics.

Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces

Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces

Modified fractional neutron point kinetics equations for finite and infinite medium of bar reactor core

Fractional order neutron point kinetics equations are generalization of the classical neutron point kinetics equations. The system of the fractional neutron point kinetics equations is treated in different context by many authors ignoring an important term related to the derivative of the reactivity. In this paper, we derive a time fractional neutron point kinetics equations (FNPK) model considering a new term for the time derivative of the reactivity. Fractional Leibenz rule is used for such derivation. The effect resulting from the additional term of the reactivity derivative is discussed. It has been found that without this additional term, the modified model will be compliant with the earlier published models for the FNPK. The proposed model of FNPK describes infinite and finite bar reactor cores according to the nonleakage probability and the geometric buckling values. We have applied an effective implicit difference approximation to solve the modified model for finite cylindrical reactor core for both average one group and six groups of delayed neutron precursors. The neutron density results with different values of fractional order for step, ramp and oscillatory reactivities are shown and compared with the classical neutron point kinetics equations.

Impact of inclined magnetic field on micropolar Casson fluid using Keller box algorithm

The present study examines the influence of an inclined magnetic field on a micropolar Casson fluid flow over a stretching sheet. Viscous dissipation effects are also taken into consideration. The governing physical problem is presented using the traditional Navier-Stokes theory. By means of the scaling group of transformation, a consequential system of equations is transformed into a set of nonlinear ordinary differential equations which are then solved using the implicit finite-difference approximation (Keller box method). The working fluid is examined for several sundry parameters graphically and in tabular form. It is observed that with an increase in inclination angle, the velocity profile decreases while temperature enhances. The Eckert number enhances flow velocity and temperature, whereas it decreases shear stress at the wall and heat transfer rate. The rheological fluid parameter contributes to the decline of velocity and temperature for weak as well as strong concentrations of micro elements.

Revised Model, with Experimental Verification, for Motor Densities in Gliding Assays

Gliding assays are used to investigate cytoskeletal polymers and motor proteins in vitro, including measurements of cytoskeletal polymer stiffness and the cooperative effects of motors. A theoretical model (Duke et al., 1995) developed to extract the biophysical quantities of interest from gliding assays has been used extensively, yet the model remains incompletely validated. In gliding assay experiments using kinesin and microtubules, we found inconsistencies in model-dependent measurements of average kinesin spacing along microtubules, a key parameter for motor cooperativity and bending stiffness interpretations. We traced these inconsistencies to an implicit approximation, an infinite processivity for kinesin. We report an updated theoretical model for gliding assays incorporating the finite processivity of kinesin, and report experimental verification of the active motor density. Deviations between densities inferred from the original model and updated model range from under 25% for short microtubules (0.5 micrometers long) to greater than four-fold for long microtubules (more than 2 micrometers long). We use these revised motor densities to measure the length-dependence of microtubule bending stiffness (persistence length).

Accurate estimation of electrostatic binding energy with Poisson-Boltzmann equation solver DelPhi program

Poisson–Boltzmann (PB) model is a widely used implicit solvent approximation in biophysical modeling because of its ability to provide accurate and reliable PB electrostatic salvation free energies ([Formula: see text] as well as electrostatic binding free energy ([Formula: see text] estimations. However, a recent study has warned that the 0.5[Formula: see text]Å grid spacing which is normally adopted can produce unacceptable errors in [Formula: see text] estimation with the solvent excluded surface (SES) (Harris RC, Boschitsch AH and Fenley MO, Influence of grid spacing in Poisson–Boltzmann equation binding energy estimation, J Chem Theory Comput 19: 3677–3685, 2013). In this work, we investigate the grid dependence of the widely used PB solver DelPhi v6.2 with molecular surface (MS) for estimating both electrostatic solvation free energies and electrostatic binding free energies. Our results indicate that, for the molecular complex and components the absolute errors of [Formula: see text] are smaller than that of [Formula: see text], and grid spacing of 0.8[Formula: see text]Å with DelPhi program ensures the accuracy and reliability of [Formula: see text]; however, the accuracy of [Formula: see text] largely relies on the order of magnitude of [Formula: see text] itself rather than that of [Formula: see text] or [Formula: see text]. Our findings suggest that grid spacing of 0.5[Formula: see text]Å is enough to produce accurate [Formula: see text] for molecules whose [Formula: see text] are large, but finer grids are needed when [Formula: see text] is very small.

Numerical solution of an inverse filtration problem

A numerical method is suggested for solving the multidimensional inverse problem of finding the flow rates of wells from given bottomhole pressures in the multidimensional flow model of a weakly compressible liquid in a poroelastic medium. Spatial approximation is based on the finite element method, which allows one to employ unstructured grids with refinements near the location of wells. Time discretization is performed with the use of implicit difference approximation. The numerical results for two- and three-dimensional test problems are presented.

Finite-difference time-domain analysis of the vibration characteristics of building structures using a dimension-reduced model

In order to accurately predict the vibration characteristics of buildings, wave-based numerical methods are effective from the viewpoint of the modeling accuracy of the physical mechanism and the detailed geometries of the simulated field. However, because of the performance of current PCs, the prediction of real-scale problems remains difficult. In order to address such problems, we herein propose a vibration simulation method for a beam-plate structure using a dimension-reduced modeling method. The target structure is modeled as a composite structure consisting of two-dimensional plate elements and one-dimensional beam elements, which are coupled based on the implicit finite-difference approximation scheme. By applying such a low-dimensional element, a faster simulation that requires less memory, as compared with a three-dimensional discretization scheme, is made available. To validate the method, the vibration characteristics obtained by the proposed scheme are compared to the measured results for model-scale and full-scale structure. The comparison of the measurement and simulation results suggest that the proposed method can be used to accurately simulate a multilayered building structure.

Compact moving least squares: An optimization framework for generating high-order compact meshless discretizations

A generalization of the optimization framework typically used in moving least squares is presented that provides high-order approximation while maintaining compact stencils and a consistent treatment of boundaries. The approach, which we refer to as compact moving least squares, resembles the capabilities of compact finite differences but requires no structure in the underlying set of nodes. An efficient collocation scheme is used to demonstrate the capabilities of the method to solve elliptic boundary value problems in strong form stably without the need for an expensive weak form. The flexibility of the approach is demonstrated by using the same framework to both solve a variety of elliptic problems and to generate implicit approximations to derivatives. Finally, an efficient preconditioner is presented for the steady Stokes equations, and the approach's efficiency and high order of accuracy is demonstrated for domains with curvi-linear boundaries.

A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential

We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular with implicit Krylov–Ritz-type approximations. An important use case for the method is the evaluation of the overlap Dirac operator in lattice Quantum Chromodynamics (QCD) at finite chemical potential, which requires the application of the sign function of a non-Hermitian matrix to some source vector. While the sign function of non-Hermitian matrices in practice cannot be efficiently approximated with source-independent polynomials or rational functions, sufficiently good approximating polynomials can still be constructed for each particular source vector. Our method allows for an efficient calculation of the derivatives of such implicit approximations with respect to the gauge field or other external parameters, which is necessary for the calculation of conserved lattice currents or the fermionic force in Hybrid Monte-Carlo or Langevin simulations. We also give an explicit deflation prescription for the case when one knows several eigenvalues and eigenvectors of the matrix being the argument of the differentiated function. We test the method for the two-sided Lanczos approximation of the finite-density overlap Dirac operator on realistic SU(3) gauge field configurations on lattices with sizes as large as 14×143 and 6×183.

A computational geometry framework for the optimisation of atom probe reconstructions

In this paper, we present pathways for improving the reconstruction of atom probe data on a coarse (>10nm) scale, based on computational geometry. We introduce a way to iteratively improve an atom probe reconstruction by adjusting it, so that certain known shape criteria are fulfilled. This is achieved by creating an implicit approximation of the reconstruction through a barycentric coordinate transform. We demonstrate the application of these techniques to the compensation of trajectory aberrations and the iterative improvement of the reconstruction of a dataset containing a grain boundary. We also present a method for obtaining a hull of the dataset in both detector and reconstruction space. This maximises data utilisation, and can be used to compensate for ion trajectory aberrations caused by residual fields in the ion flight path through a ‘master curve’ and correct for overall shape deviations in the data.

Multilevel hybrid split-step implicit tau-leap

In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (Explicit-TL) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL) at levels where the SSI-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

Implementing high performance voltammetry simulation using the implicit parallel algorithm

The numerical simulation of voltammetry and amperometry at a microdisc electrode is implemented with a parallel algorithm using GPU (CUDA in C++) and adapting the implicit finite difference approximation with the ADI method. This leads to high performance simulation of the current response whilst maintaining the high standard of accuracy required for such electrochemical systems. We gained a maximum speed-up of 20 times with a standard graphic card specifications (Quadro K4000) in comparison to a single core of 3.2GHz CPU. The simplified allocation and data flow allows an accessible code which produces a starting point for researchers to manipulate the code to study new problems in electrochemical and surface reaction systems that require high performance simulation.

Alternating triangular schemes for convection–diffusion problems

Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams.

Modeling of nonlocal damage-plasticity in beams using isogeometric analysis

A nonlocal damage-plastic Euler–Bernoulli beam model is proposed based on resultant damage-plasticity together with implicit approximations of the nonlocal integral operators. Isogeometric concept is used to approximate the higher order gradient terms, and numerical algorithms and consistent tangents are derived. Test cases are presented to show the important features of the proposed nonlocal beam element. Numerical results show that the proposed nonlocal beam is successfully able to preserve the objectivity of the results and a meaningful convergence is achieved with decreasing mesh size. In addition, various softening responses associated with localized damage mechanisms – including snap-back and snap-through responses – are obtained.

Numerical technology for landslide tsunami simulations based on Navier-Stokes equations

The paper presents an integral technology simulating all phases of a landslide-driven tsunami. The technology is based on the numerical solution of a set of Navier-Stokes equations for multi-phase flows. The numerical algorithm uses a fully implicit approximation method, in which the equations of continuity and momentum conservation are coupled through implicit summands of pressure gradient and mass flow. The method we propose removes strict constraints on the timestep and makes it possible to simulate tsunami propagation to arbitrarily large distances. The landslide origin is simulated as an individual phase being a Newtonian fluid with its own density and viscosity and separated from the water and air phases by an interface. The paper presents the basic equation discretization formulas and expressions for coefficients and describes the main steps of the computational procedure. To enable simulations of tsunami propagation across wide water areas, we propose a parallel technology implementation algorithm that employs an algebraic multi-grid method. The multi-grid method implementation is based on the global and cascade collection algorithms, which impose no limitations on the paralleling scale and make this technology applicable to petascale systems. We demonstrate the possibility of simulating all phases of a landslide-driven tsunami, including its generation, propagation and uprush. The technology has been verified against the problems supported by experimental data. The paper describes the mechanism of incorporating bathymetric data to simulate tsunamis in real water areas of the world ocean. Results of comparison with the non-linear dispersion theory, which demonstrated good agreement, are presented for the case of a historical tsunami of volcanic origin on the Montserrat island in the Caribbean Sea.