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Conventional wisdom suggests that high-order finite-difference methods are more efficient than high-order discontinuous spectral-element methods on smooth meshes, but less efficient as the mesh becomes increasingly distorted because of a significant loss of accuracy on such meshes. This paper investigates the influence of mesh distortion on the relative efficiency of different implementations of generalized summation-by-parts (GSBP) methods, with emphasis on comparing finite-difference and discontinuous spectral-element approaches. These include discretizations built using classical finite-difference SBP operators, with and without optimized boundary closures, as well as both Legendre-Gauss and Legendre-Gauss-Lobatto operators. The traditionally finite-difference operators are also applied as discontinuous spectral-element operators by selecting a fixed number of nodes per element and performing mesh refinement by increasing the number of elements rather than the number of mesh nodes. Using the linear convection equation and nonlinear Euler equations as models, solutions are obtained on meshes with different types and severity of distortion. Contrary to expectation, the results show that finite-difference implementations are no more sensitive to mesh distortion than discontinuous spectral-element implementations, maintaining their relative efficiency in most cases. The results also show that the operators of Mattsson et al. (J Comput Phys 264:91–111, 2014) with optimized boundary operators are often the most efficient for a given implementation strategy (finite-difference or discontinuous spectral-element). While their accuracy as finite-difference operators might be expected, their superior accuracy to LG and LGL nodal distributions when implemented as discontinuous spectral-element operators is not well known. [ABSTRACT FROM AUTHOR]

We propose an efficient implementation of the numerical tensor-train (TT) based algorithm solving the multicomponent coagulation equation preserving the nonnegativeness of solution. Unnatural negative elements in the constructed approximation arise due to the errors of the low-rank decomposition and discretization scheme. In this work, we propose to apply the rank-one corrections in the TT-format proportional to the minimal negative element. Such an element can be found via application of the global optimization methods that can be fully implemented within efficient operations in the tensor train format. We incorporate this trick into the time-integration scheme for the multicomponent coagulation equation and also use it for post-processing of the stationary solution for the problem with the source of particles. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary. [ABSTRACT FROM AUTHOR]

Objectives. A finite-difference computational algorithm is proposed for solving a mixed boundary-value problem for the Poisson equation given in two-dimensional irregular domains.Methods. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain.The obtained results are mapped on non-uniform boundary-fitted difference grid in the physical domain.Results. The second order approximations of mixed Neumann-Dirichlet boundary conditions for the Poisson equation in the space of generalized curvilinear coordinate are constructed. To increase the order of Neumann condition approximations, an approximation of the Poisson equation on the boundary of the domain is used.Conclusions. To solve a mixed boundary value problem for the Poisson equation in two-dimensional irregular domains, the computational algorithm of second-order accuracy is constructed. The generalized curvilinear coordinates are used. The results of numerical experiments, which confirm the second order accuracy of the computational algorithm, are presented.

A numerico-analytical algorithm for estimating the rounding errors in the uniform metric is developed. Their boundedness is established over the entire range of calculating the current-voltage characteristics of long Josephson junctions using the proposed second-order scheme. For a system of two difference equations as an example, it is shown how the growth rate of rounding errors in the uniform metric can be analyzed numerically in the case of a power-law instability. In addition, estimates are obtained for the growth rate of the rounding errors in the uniform metric for the third-order Rusanov scheme. The calculations were carried out on the Govorun supercomputer using the REDUCE system. [ABSTRACT FROM AUTHOR]

Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to many reactions. As a result, their mathematical models depend on many parameters. In applications, some of the model parameters may be unknown, so their values need to be estimated from the experimental data. However, the problem of parameter value inference can be quite challenging, especially in the stochastic setting. To estimate accurately the values of a subset of parameters, the system should be sensitive with respect to variations in each of these parameters and they should not be correlated. In this paper, we propose a technique for detecting collinearity among models' parameters and we apply this method for selecting subsets of parameters that can be estimated from the available data. The analysis relies on finite-difference sensitivity estimations and the singular value decomposition of the sensitivity matrix. We illustrated the advantages of the proposed method by successfully testing it on several models of biochemical systems of practical interest. [ABSTRACT FROM AUTHOR]

This work is devoted to increasing the computational efficiency of numerical methods for the one-way Helmholtz Equation (higher-order parabolic equation) in a heterogeneous underwater environment. The finite-difference rational Padé approximation of the propagation operator is considered, whose artificial computational parameters are the grid cell sizes and reference sound speed. The relationship between the parameters of the propagation medium and the artificial computational parameters is established. An optimized method for automatic determination of the artificial computational parameters is proposed. The optimization method makes it possible to account for any propagation angle and arbitrary variations in refractive index. The numerical simulation results confirm the adequacy and efficiency of the proposed approach. Automating the selection process of the computational parameters makes it possible to eliminate human errors and avoid excessive consumption of computational resources. [ABSTRACT FROM AUTHOR]

In this study, an efficient and accurate staggered-grid finite-difference time-domain method to solve the two-dimensional (2-D) first-order stress–velocity elastic-wave equation is proposed. In the conventional implementation of the staggered-grid finite-difference (SGFD) method, the same SGFD operator is used to approximate the spatial derivatives. However, we propose a numerical method based on the mixed SGFD operators that are more efficient but similar in accuracy when compared with a uniform SGFD operator. We refer to the proposed method as the nonbalanced SGFD numerical scheme that combines the high-order SGFD operators with the second-order SGFD operators. The suitability of the proposed scheme is verified by dispersion analysis. Through SGFD modeling and reverse-time migration examples, we demonstrate that the proposed nonbalanced scheme offers a similar level of accuracy with a lower computational cost compared with the time-consuming conventional SGFD method.

Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to many reactions. As a result, their mathematical models depend on many parameters. In applications, some of the model parameters may be unknown, so their values need to be estimated from the experimental data. However, the problem of parameter value inference can be quite challenging, especially in the stochastic setting. To estimate accurately the values of a subset of parameters, the system should be sensitive with respect to variations in each of these parameters and they should not be correlated. In this paper, we propose a technique for detecting collinearity among models’ parameters and we apply this method for selecting subsets of parameters that can be estimated from the available data. The analysis relies on finite-difference sensitivity estimations and the singular value decomposition of the sensitivity matrix. We illustrated the advantages of the proposed method by successfully testing it on several models of biochemical systems of practical interest.

The aim of this paper is to develop an implicit–explicit (IMEX) scheme for solving the 2-dimensional (2-D) partial integro-differential equations with spatial delays arising in option pricing under the hard-to-borrow jump-diffusion models. First, a new second-order accurate numerical integration scheme that combines a mesh-dependent expansion and the trapezoidal rule is proposed to handle the integral-delayed term. Then, the IMEX scheme discretizes the integral-delayed term explicitly and the other terms implicitly. The second-order convergence rates for space and time are proved. Numerical examples are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

We consider moving singular points of systems of ordinary differential equations. A review of Painlevs results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Alshina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlev property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.

This work is devoted to increasing the computational efficiency of numerical methods for the one-way Helmholtz Equation (higher-order parabolic equation) in a heterogeneous underwater environment. The finite-difference rational Padé approximation of the propagation operator is considered, whose artificial computational parameters are the grid cell sizes and reference sound speed. The relationship between the parameters of the propagation medium and the artificial computational parameters is established. An optimized method for automatic determination of the artificial computational parameters is proposed. The optimization method makes it possible to account for any propagation angle and arbitrary variations in refractive index. The numerical simulation results confirm the adequacy and efficiency of the proposed approach. Automating the selection process of the computational parameters makes it possible to eliminate human errors and avoid excessive consumption of computational resources.

Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement. [ABSTRACT FROM AUTHOR]

Time-lapse resistivity tomography bring valuable information on the physical changes occurring inside a geological reservoir. In this study, resistivity monitoring from controlled source electromagnetics (CSEM) data is investigated through synthetic and real data. We present three different schemes currently used to perform time-lapse inversions and compare these three methods: parallel, sequential and double difference. We demonstrate on synthetic tests that double difference scheme is the best way to perform time-lapse inversion when the survey parameters are fixed between the different time-lapse acquisitions. We show that double difference inversion allows to remove the imprint of correlated noise distortions, static shifts and most of the non-linearity of the inversion process including numerical noise and acquisition footprint. It also appears that this approach is robust against the baseline resistivity model quality, and even a rough starting resistivity model built from borehole logs or basic geological knowledge can be sufficient to map the time-lapse changes at their right positions. We perform these comparisons with real land time-lapse CSEM data acquired one year apart over the Reykjanes geothermal field. [ABSTRACT FROM AUTHOR]

We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced. [ABSTRACT FROM AUTHOR]

A Finite-Difference Time-Domain (FDTD) simulation of broadband electromagnetic metasurfaces based on direct incorporation of Generalized Sheet Transition Conditions (GSTCs) into a conventional Yee-cell region has been proposed for arbitrary wave excitations. This is achieved by inserting a zero thickness metasurface inside bulk nodes of the Yee-cell region, giving rise to three distinct cell configurations - Symmetric Cell (SC), Asymmetric Cell (AC) and Tight Asymmetric Cell (TAC). In addition, the metasurface is modelled using electric and magnetic surface susceptibilities exhibiting a broadband Lorentzian response. As a result, the proposed model guarantees a physical and causal response from the metasurface. Several full-wave results are shown and compared with analytical Fourier propagation methods showing excellent results for both 1D and 2D field simulations. It is found that the TAC provides the fastest convergence among the three methods with minimum error.

The aim of the paper is to estimate the minimum appropriate number of nodes on a uniform grid (maximum integration step) to obtain a given accuracy for the finite-difference Runge-Kutta methods of the first and second orders of accuracy for the Dahlquist model equation.The error of finite-difference methods is analytically investigated by explicit comparing the values of the exact solutions of the differential and difference Cauchy problems in the nodes of a uniform grid in modulus, and the global error is determined by the maximum of the modules of the local errors on the selected grid. The estimates of the global error are obtained from the inequalities based on the expansions of the functions of the exponent and the logarithm in the Taylor and Mercator series, and clearly depend on the number of nodes of the uniform grid.The bottom of the number of nodes of the uniform grid that is required to have the desirable accuracy to solve the Cauchy problem by above methods is obtained.The obtained estimate of the global error of the direct Euler method for the Dahlquist model equation substantially refines the similar estimate from the paper (Hairer E., and Lubich C. Numerical Solution of Ordinary Differential Equations) and enables us to use an integration step of 1.7 times more in value, keeping the given approximation accuracy.The accuracy order of the finite-difference schemes in the theory of numerical methods for integrating differential equations provides a relationship between the global error of the method and the integration step, however, it does not allow us to directly express the approximation accuracy on the given grid, and therefore, an optimal integration step is most often determined experimentally. The paper studies such a relationship explicitly as a model example and shows one of the possible ways to obtain analytical estimates of the integration step for a given approximation accuracy.A direct study of the global error of finite-difference schemes is important in problems where a trade-off between the approximation accuracy and the complexity (amount of computation) is of importance when the number of grid nodes matters. In this regard, it is of interest to extend similar studies of error estimation to the other finite-difference schemes, namely Runge-Kutta methods of higher orders of accuracy and multistep methods.The results obtained can be useful for solving the tasks of computer modeling and computer-based learning.

Sucker-rod pumping wells can be either vertical or directional. Over time, research efforts on the functioning of vertical wells led to a well-established set of mathematical models and practical tools. When it comes to directional wells, however, no general agreement has been reached, and the topic remains in active discussion. This paper revisits, extends, implements and optimizes an overlooked model, initially devised in 1995, whose computational complexity resulted in long processing times that stymied its adoption. This model fully utilizes the 3D trajectory of the rod string, allowing for the use of two viscous friction models and proposing its own formulation for downhole boundary conditions. The resulting model can be used to efficiently simulate the dynamic behavior of directional sucker-rod pumping wells taking into account the fluid flow inside the rod-tubing annulus. We present and analyze a serial and a parallel software implementation of this CPU-intensive model based on an explicit finite-difference method. We also describe our contributions to the accuracy and performance of the original model and software implementation. A rough approximation shows that the proposed serial version is about 200 times faster than the legacy original code, if we were to run the latter in a modern processor. On top of that our parallel implementation achieved a 6.5 × speedup over the serial version in a shared-memory system, making it a suitable tool for well design and optimization. The research contributes to the discussions on mathematical modeling of directional sucker-rod pumping wells, and illustrates how performance-focused techniques can enable the effective use of computationally demanding models to facilitate further refinements and applications. [ABSTRACT FROM AUTHOR]

This study puts forward construction of an efficient nonpolynomial twin parameter cubic spline-based numerical scheme for approximations to the solution of heat transfer and defection in cables problems represented as system of second-order boundary-value problems. The introduction of an additional parameter in trigonometric part of nonpolynomial cubic spline makes this scheme a better one as compared to other existing numerical methods. The Icing on the cake is the applicability of the proposed scheme for unequal step size. The present algorithm gives better approximations in comparison to other spline, collocation, and finite-difference methods. The convergence analysis of the proposed algorithm is talked about to make a strong foundation to the proposed algorithm. Practical usefulness of the proposed method is illustrated through numerical examples. [ABSTRACT FROM AUTHOR]

This article proposes an efficient finite-difference time-domain (FDTD) technique for determining electromagnetic fields interacting with a finite-sized 2-D and 3-D periodic structures. The technique combines periodic boundary conditions—modeling fields away from the edges of the structure—with independent simulations of fields near the edges of the structure. It is shown that this algorithm efficiently determines the size of a periodic structure necessary for fields to converge to the infinitely periodic case. Numerical validations of the technique illustrate the savings concomitant with the algorithm. [ABSTRACT FROM AUTHOR]

The results of numerical modelling including computer-generated graphics of deformation within a viscouse incompressible fluid layer at the transfer process onto a substrate between contacted cylinders are presented. The modelling is performed using developed algorithm of numerical calculation of Navie-Stocks evaluations for the viscouse incompressible fluidon a two-dimensional mesh grid by finite-difference methods with additional computation of moving borders taking to account fluid splitting, fluid microdrops formation and contacting surfaces deformation due to pressure difference. A laminar type of the viscouse incompressible fluidflow at a surfaces contact point incase of a permanent cylinder rotation is considered. It is noted that the programm implementation of the developed algorithm as exemplified by a printing system allows to automate ink transаer coefficient estimation and also to predict printing system parameters.

In this paper, we present two different approaches for constructing reduced‐order models (ROMs) for the two‐dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear‐quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem. [ABSTRACT FROM AUTHOR]

Density Matrix Formalism using quantum methods has been used for determining the channel density of dual gate ultra-thin MOSFETs. The results obtained from the quantum methods have been compared with the semi-classical methods. This paper discusses in detail the simulation methods using self-consistent schemes and the discretization procedures for constructing the Hamiltonian Matrix for a dual gate MOSFET consisting of oxide/semiconductor/oxide interface and the self-consistent procedure involving the discretization of Poisson's equation for satisfying the charge neutrality condition in the channel of different thicknesses. Under quantum methods, the channel densities are determined from the diagonal elements of the density matrix. This successfully simulates the size quantization effect for thin channels. For semi-classical methods, the Fermi–Dirac Integral function is used for the determination of the channel density. For thin channels, the channel density strongly varies with the values of the effective masses. This variation is simulated when we use Quantum methods. The channel density also varies with the asymmetric gate bias and this variation is more for thicker channels where the electrons get accumulated near the oxide/semiconductor interface. All the calculations are performed at room temperature (300 K). [ABSTRACT FROM AUTHOR]

Wireless networks of the next generation should optimally adapt themselves to the operating environment in order to meet the requirements of a significantly higher throughput at a lower energy consumption and at increased utilization of radio resources. The current methods used for the radio environment modelling are insufficiently accurate. Older empirical models can no longer be used to model channel spatial and temporal characteristics because of their inadequacy. The widespread deterministic modelling based on geometric optics significantly deviates from the radio frequency diffraction and scattering behavior. On the other hand, the numerical finite-difference techniques based on the Maxwell’s partial differential equations of electrodynamics do provide the required level of details but are limited to rather small computation domains. However, the increasing number of the access points and the associated reduction in the wireless cell reach suggest to use numerical techniques as a viable alternative to the ray optical modelling of the telecommunication channels. The paper reviews the current challenges and approaches to solving the large-scale telecommunication problems by using the numerical finite-difference techniques and proposes to adapt them because of their considerable acceptability. The techniques complemented with the use of hardware accelerators can become an efficient accuracy-improved alternative of the deterministic models. [ABSTRACT FROM AUTHOR]

Results obtained in recent years concerning necessary and sufficient conditions for the convergence (at a given rate) of approximation methods for solutions of irregular operator equations are overviewed. The exposition is given in the context of classical direct and converse theorems of approximation theory. Due to the proximity of the resulting necessary and sufficient conditions to each other, the solutions on which a certain convergence rate of the methods is reached can be characterized nearly completely. The problems under consideration include irregular linear and nonlinear operator equations and ill-posed Cauchy problems for first- and second-order differential operator equations. Procedures for stable approximation of solutions of general irregular linear equations, classes of finite-difference regularization methods and the quasi-reversibility method for ill-posed Cauchy problems, and the class of iteratively regularized Gauss–Newton type methods for irregular nonlinear operator equations are examined. [ABSTRACT FROM AUTHOR]

In this paper, we construct finite-difference approximations of optimal-control problems involving non-self-adjoint convection-diffusion elliptic equations with discontinuous coefficients and states and examine the convergence of these approximations. Control functions in these problems are the coefficients of the convective-transfer operator in the equation of state and its right-hand side. We study the well-posedness of problems considered. For finite-difference approximations, we obtain estimates of the exactness by the state and the convergence rate by the functional and prove the weak convergence by the control. In addition, we regularize approximations in the Tikhonov sense. [ABSTRACT FROM AUTHOR]

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

The second part of paper is devoted to final study of three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Specificity of this solution has been shown by different examples, which are related to the hexagonal grid. Also the next statements that are mentioned in the first part have been proved: the matter conservation law and the offensive effect of excessive hexagonal symmetry. From the point of CA view diffusion equation is the most important. While solving of diffusion equation at the infinite time interval we can find solution of boundary value problem of Laplace equation and if we introduce vector-variable we will solve wave equation (at least, for scalar). The critical requirement for the sampling of the boundary conditions for CA-cells has been shown during the solving of problem of circular membrane vibrations with Neumann boundary conditions. CA-calculations using the simple scheme and Margolus rotary-block mechanism were compared for the quasione-dimensional problem "diffusion in the half-space". During the solving of mixed task of circular membrane vibration with the fixed ends in a classical case it has been shown that the simultaneous application of the Crank-Nicholson method and taking into account of the second-order terms is allowed to avoid the effect of excessive hexagonal symmetry that was studied for a simple scheme. By the example of the centrally symmetric Neumann problem a new method of spatial derivatives introducing into the postfix CA procedure, which is reflecting the time derivatives (on the base of the continuity equation) was demonstrated. The value of the constant that is related to these derivatives has been empirically found in the case of central symmetry. The low rate of convergence and accuracy that limited within the boundaries of the sample, in contrary to the formal precision of the method (4-th order), prevents the using of the CAmethods for such problems. We recommend using multigrid method. During the solving of the quasi-diffusion equations (two-dimensional CA) it was showing that the rotary-block mechanism of CA (Margolus mechanism) is more effective than simple CA.

The paper has methodical character; it is devoted to three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Special attention was payed to the matter conservation law and the offensive effect of excessive hexagonal symmetry. It has been shown that in contrary to finite-difference approach, in spite of terminological equivalence of CA local transition function to the pattern of computing double layer explicit method, CA approach contains the replacement of matrix technique by iterative ones (for instance, sweep method for three diagonal matrixes). This suggests that discretization of boundary conditions for CA-cells needs more rigid conditions. The correct local transition function (LTF) of the boundary cells, which is valid at least for the boundaries of the rectangular and circular shapes have been firstly proposed and empirically given for the hexagonal grid and the conservative boundary conditions. The idea of LTF separation into «internal», «boundary» and «postfix» have been proposed. By the example of this problem the value of the Courant-Levy constant was re-evaluated as the CA convergence speed ratio to the solution, which is given at a fixed time, and to the rate of the solution change over time.

Plasma diagnostics is a topic of great interest in the physics and engineering community because the monitoring of plasma parameters plays a fundamental role in the development and optimization of plasma reactors. Towards this aim, microwave diagnostics, such as reflectometric, interferometric, and polarimetric techniques, can represent effective means. Besides the above, microwave imaging profilometry (MIP) may allow the obtaining of tomographic, i.e., volumetric, information of plasma that could overcome some intrinsic limitations of the standard non-invasive diagnostic approaches. However, pursuing MIP is not an easy task due to plasma's electromagnetic features, which strongly depend on the working frequency, angle of incidence, polarization, etc., as well as on the need for making diagnostics in both large (meter-sized) and small (centimeter-sized) reactors. Furthermore, these latter represent extremely harsh environments, wherein different systems and equipment need to coexist to guarantee their functionality. Specifically, MIP entails solution of an inverse scattering problem, which is non-linear and ill-posed, and, in addition, in the one-dimensional case, is also severely limited in terms of achievable reconstruction accuracy and resolution. In this contribution, we address microwave inverse profiling of plasma assuming a high-frequency probing regime when magnetically confined plasma can be approximated as both an isotropic and weak penetrable medium. To this aim, we adopt a finite-difference frequency-domain (FDFD) formulation which allows dealing with non-homogeneous backgrounds introduced by unavoidable presence of plasma reactors. [ABSTRACT FROM AUTHOR]

It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103-125, 1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg-de Vries equation to investigate the shape of the moving body capable of generating the highest first upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norms and compact supports. We establish analytically the existence of a maximizer saturating the norm constraint, derive the gradient of the functional, and then implement numerically an optimization algorithm yielding the desired optimal shape. [ABSTRACT FROM AUTHOR]

Numerous geological processes are governed by thermal and mechanical interactions. In particular, tectonic processes such as ductile strain localization can be induced by the intrinsic coupling that exists between deformation, energy and rheology. To investigate this thermomechanical feedback, we have designed 2-D codes that are based on an implicit finite-difference discretization. The direct-iterative method relies on a classical Newton iteration cycle and requires assembly of sparse matrices, while the pseudo-transient method uses pseudo-time integration and is matrix-free. We show that both methods are able to capture thermomechanical instabilities when applied to model thermally activated shear localization; they exhibit similar temporal evolution and deliver coherent results both in terms of nonlinear accuracy and conservativeness. The pseudo-transient method is an attractive alternative, since it can deliver similar accuracy to a standard direct-iterative method but is based on a much simpler algorithm and enables high-resolution simulations in 3-D. We systematically investigate the dimensionless parameters controlling 2-D shear localization and model shear zone propagation in 3-D using the pseudo-transient method. Code examples based on the pseudo-transient and direct-iterative methods are part of the M2Di routines (Räss et al. 2017) and can be downloaded from Bitbucket and the Swiss Geocomputing Centre website. [ABSTRACT FROM AUTHOR]

Diffusion of the pulse magnetic field into the metal of the toroidal magnetic core wound by a transformer steel strap is considered. The 1-D numerical model has been developed, which takes into account the viscous-type dynamic losses of magnetization, as well as the eddy currents. The parameter of viscous-type dynamic losses of magnetization was measured on samples of cold-rolled anisotropic transformer steel. The processes of magnetic field propagation in the strap, when the transformer is switched to the voltage source, have been analyzed, and the iteration algorithm of calculation of magnetic field in the strap has been developed. The experimental data have confirmed the obtained calculation results. The problems of application of transformer steel in pulse transformers are considered. [ABSTRACT FROM AUTHOR]

A two-dimensional model of the Frederiks effect is used to investigate the static electric field effect on the orientation of the nematic liquid crystal (LC) director in a side-electrode cell. The solutions are obtained by the standard finite-difference methods. The programs for the numerical solution of a two-dimensional parabolic partial differential equation are developed both in FORTRAN and C/C++. The Frederiks transition threshold for the central part of the cell and the dependences of the director’s orientation distribution on a high electric field are obtained. The results of the calculation are compared with the experimental data. [ABSTRACT FROM AUTHOR]

The acoustic approximation for anisotropic media is widely used in current industry imaging and inversion algorithms mainly because P waves constitute the majority of the energy recorded in seismic exploration. The resulting acoustic formulae tend to be simpler, resulting in more efficient implementations, and depend on fewer medium parameters. However, conventional solutions of the acoustic wave equation with higher-order derivatives suffer from shear wave artefacts. Thus, we derive a new acoustic wave equation for wave propagation in transversely isotropic (TI) media, which is based on a partially separable approximation of the dispersion relation for TI media and free of shear wave artefacts. Even though our resulting equation is not a partial differential equation, it is still a linear equation. Thus, we propose to implement this equation efficiently by combining the finite difference approximation with spectral evaluation of the space-independent parts. The resulting algorithm provides solutions without the constraint δ ≥ δ. Numerical tests demonstrate the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

A new multipole perfectly matched layer (PML) formulation is presented. Based on the stretched-coordinate approach, the formulation that utilizes a recursive integration concept in its development, introduces a PML stretching function that is created as the sum of any given number of complex-frequency shifted (CFS) constituent poles. Complete formulae for up to a three-pole formulation, to facilitate its implementation in finite-difference time-domain codes, are developed. The performance of this new multipole formulation compares favorably with existing higher order PMLs that instead utilize stretching functions that are developed as the product of elementary CFS constituent poles. It is argued that the optimization of the new multipole PML (MPML) could be more straightforward when compared to that of a higher order PML due to the absence of extra terms generated by the process of multiplication used in the development of the overall PML stretching function in higher order PMLs. The new MPML is found to perform very well when compared to standard CFS-PMLs requiring equivalent computational resources. [ABSTRACT FROM AUTHOR]

A technique for finding numerical similarity solutions to an initial boundary value problem (IBVP) for generalized K(m, n) equations is described. The equation under consideration is nonlinear and has variable coefficients. The original problem is transformed with the aid of Lie symmetries to an initial value problem (IVP) for a nonlinear third-order ordinary differential equation. The existence and uniqueness of the solution are examined, and the problem is consequently solved with the aid of a finite-difference scheme for various values of the governing parameters. In lieu of an exact symbolic solution, the scheme is validated by comparing the numerical solutions with the approximate analytic solutions obtained with the aid of the method of successive approximations in their region of validity. The accuracy, efficiency, and consistency of the scheme are demonstrated. Numerical solutions to the original initial boundary value problem are constructed for selected parameter values with the aid of the transforms. The qualitative behavior of the solutions as a function of the governing parameters is analyzed, and it is found that the examined IBVPs for generalized K(m, n) equations with variable coefficients that are functions of time, do not admit solitary wave or compacton solutions. [ABSTRACT FROM AUTHOR]

The Lebedev grid is an alternative to Yee grid where, because the discretized fields lie at the same location in space, there is no need for further field interpolations when modeling anisotropic materials. In this letter, we focus on accurately implementing perfect electric conductors with planar interfaces, as well as 90° and 270° corners. Simulations are provided to demonstrate the overall improvements and also the second-order accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

Plasma diagnostics is a topic of great interest in the physics and engineering community because the monitoring of plasma parameters plays a fundamental role in the development and optimization of plasma reactors. Towards this aim, microwave diagnostics, such as reflectometric, interferometric, and polarimetric techniques, can represent effective means. Besides the above, microwave imaging profilometry (MIP) may allow the obtaining of tomographic, i.e., volumetric, information of plasma that could overcome some intrinsic limitations of the standard non-invasive diagnostic approaches. However, pursuing MIP is not an easy task due to plasma’s electromagnetic features, which strongly depend on the working frequency, angle of incidence, polarization, etc., as well as on the need for making diagnostics in both large (meter-sized) and small (centimeter-sized) reactors. Furthermore, these latter represent extremely harsh environments, wherein different systems and equipment need to coexist to guarantee their functionality. Specifically, MIP entails solution of an inverse scattering problem, which is non-linear and ill-posed, and, in addition, in the one-dimensional case, is also severely limited in terms of achievable reconstruction accuracy and resolution. In this contribution, we address microwave inverse profiling of plasma assuming a high-frequency probing regime when magnetically confined plasma can be approximated as both an isotropic and weak penetrable medium. To this aim, we adopt a finite-difference frequency-domain (FDFD) formulation which allows dealing with non-homogeneous backgrounds introduced by unavoidable presence of plasma reactors.