Search

In this paper, we consider the numerical implementation of the 2D wave equation in isotropic-heterogeneous media. The stability analysis of the scheme using the von Neumann stability method has been studied. We conducted a study on modeling the propagation of acoustic waves in a heterogeneous medium and performed numerical simulations in various heterogeneous media at different time steps. Developed parallel code using Compute Unified Device Architecture (CUDA) technology and tested on domains of various sizes. Performance analysis showed that our parallel approach showed significant speedup compared to sequential code on the Central Processing Unit (CPU). The proposed parallel visualization simulator can be an important tool for numerous wave control systems in engineering practice. [ABSTRACT FROM AUTHOR]

In this paper, the Crank-Nicholson method is applied to solve the one-dimensional nonlinear Burgers’ equation in warm, dusty plasmas with dust charge variation. After obtaining numerical results, a thorough analysis is conducted and compared against analytical solutions. On the basis of the comparison, it is evident that the numerical results obtained from the analysis are in good agreement with the analytical solution. The error between the analytical and numerical solutions of the Burgers’ equation is calculated by two error norms, namely L2 and L∞. A Von-Neumann stability analysis is performed on the present method, and it is found to be unconditionally stable according to the Von-Neumann analysis.

In this work, the pseudoparabolic problem of the fourth order is investigated to identify the time -dependent potential term under periodic conditions, namely, the integral condition and overdetermination condition. The existence and uniqueness of the solution to the inverse problem are provided. The proposed method involves discretizing the pseudoparabolic equation by using a finite difference scheme, and an iterative optimization algorithm to resolve the inverse problem which views as a nonlinear least-square minimization. The optimization algorithm aims to minimize the difference between the numerical computing solution and the measured data. Tikhonov’s regularization method is also applied to gain stable results. Two examples are introduced to explain the reliability of the proposed scheme. Finally, the results showed that the time dependent potential terms are successfully reconstructed, stable and accurate, even in inclusion of noise. [ABSTRACT FROM AUTHOR]

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method. [ABSTRACT FROM AUTHOR]

In this paper, the Crank-Nicholson method is applied to solve the one-dimensional nonlinear Burgers' equation in warm, dusty plasmas with dust charge variation. After obtaining numerical results, a thorough analysis is conducted and compared against analytical solutions. On the basis of the comparison, it is evident that the numerical results obtained from the analysis are in good agreement with the analytical solution. The error between the analytical and numerical solutions of the Burgers' equation is calculated by two error norms, namely L2 and L∞. A Von-Neumann stability analysis is performed on the present method, and it is found to be unconditionally stable according to the Von-Neumann analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the numerical solutions of time fractional Burger’s and coupled Burgers’ equations are obtained using the parametric quintic spline method with a local truncation error of order eight in distance direction. Additionally, the von Neumann method was utilized for studying the stability analysis of the present method. Finally, to show the accuracy of this method, some examples with different cases for Burger’s and coupled Burgers’ equations are presented and their results are compared with the previous methods.

This paper presents a comprehensive study of the numerical simulation of the one-dimensional modified Burgers' equation in dusty plasmas. The reductive perturbation method is employed to derive the equation, and a numerical solution is obtained using the explicit finite difference technique. The obtained results are extensively compared with analytical solutions, demonstrating a high level of agreement, particularly for lower values of the dissipation coefficient. The accuracy and efficiency of the technique are evaluated based on the absolute error. Additionally, the accuracy and effectiveness of the technique are assessed by plotting L2 and L∞ error graphs. The technique's reliability is further confirmed through von Neumann stability analysis, which indicates that the technique is conditionally stable. Overall, the study concludes that the proposed technique is successful and dependable for numerically simulating the modified Burgers' equation in dusty plasmas.

A numerical method for solving one-dimensional (1D) parabolic convection–diffusion equation is provided. We consider the finite difference formulas with five points to obtain a numerical method. The proposed method converts the given equation, domain, and time interval into a discrete form. The numerical values of the solution are approximated by solving algebraic equations containing finite differences and values at these discrete points. The consistency, stability and convergence are investigated. On the other hand, some numerical examples illustrate the validity and applicability of the method. Finally, the numerical results are compared with the finite difference scheme’s three points.

This paper examines the finding of spacewise dependent heat source function in pseudoparabolic equation with initial and homogeneous Dirichlet boundary conditions, as well as the final time value / integral specification as additional conditions that ensure the uniqueness solvability of the inverse problem. However, the problem remains ill-posed because tiny perturbations in input data cause huge errors in outputs. Thus, we employ Tikhonov’s regularization method to restore this instability. In order to choose the best regularization parameter, we employ L-curve method. On the other hand, the direct (forward) problem is solved by a finite difference scheme while the inverse one is reformulated as an optimization problem. The later problem is accomplished by employing lsqnolin subroutine from MATLAB. Two test examples are presented to show the efficiency and accuracy of the employed method by including many noises level and various regularization parameters. [ABSTRACT FROM AUTHOR]

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method.

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers' equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solutions of time fractional Burger's and coupled Burgers' equations are obtained using the parametric quintic spline method with a local truncation error of order eight in distance direction. Additionally, the von Neumann method was utilized for studying the stability analysis of the present method. Finally, to show the accuracy of this method, some examples with different cases for Burger's and coupled Burgers' equations are presented and their results are compared with the previous methods. [ABSTRACT FROM AUTHOR]

A numerical method for solving one-dimensional (1D) parabolic convection–diffusion equation is provided. We consider the finite difference formulas with five points to obtain a numerical method. The proposed method converts the given equation, domain, and time interval into a discrete form. The numerical values of the solution are approximated by solving algebraic equations containing finite differences and values at these discrete points. The consistency, stability and convergence are investigated. On the other hand, some numerical examples illustrate the validity and applicability of the method. Finally, the numerical results are compared with the finite difference scheme's three points. [ABSTRACT FROM AUTHOR]

B-spline is extensively used for the solution of many physical models appearing in the fields of plasma physics, fluid mechanics, atmosphere-ocean dynamics and many other disciplines. In this article, Riesz space fractional PDEs (RSF-PDEs) in two forms are solved by using hybrid B-spline collocation method (HBCM). In the given methodology, RSF-PDEs are discretized into the system of algebraic linear equations by using hybrid B-spline basis function. The resultant system is solved by a numerical technique. The Von Neumann stability analysis method is used for analyzing the stability of proposed method. Numerical experiments are conducted to illustrate the accuracy of proposed method, by representing the results graphically and numerically for different values of fractional parameters.

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation. [ABSTRACT FROM AUTHOR]

This work aims at presenting a new numerical solution to a nonlinear, one-dimensional problem of heat wave propagation in a thick slab of a rigid thermal conductor. The model predicts dependence of second sound velocity on temperature and heat flux. For this, an unconditionally stable numerical scheme is constructed using a kind of weighted average nonstandard finite difference discretization. Stability analysis of this scheme is studied by von Neumann technique, and its accuracy is proved. Numerical simulations are given for all quantities of physical interest to confirm the reliability of the proposed method. The model and the numerical results shed light on characteristics of heat wave propagation within the theory of extended thermodynamics. [ABSTRACT FROM AUTHOR]

Abstract A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In this work, an application of fifth degree basis spline functions is presented for a numerical investigation of the Kuramoto–Sivashinsky equation. The finite forward difference formula is used for temporal integration, whereas the basis splines, together with a new approximation for fourth order spatial derivative, are brought into play for discretization in space direction. In order to corroborate the presented numerical algorithm, some test problems are considered and the computational results are compared with existing methods.

The present study details the generalization of a stability condition for the semi-implicit formulation of the one-dimensional impurity transport equation for tokamak plasmas in terms of the magnetic flux surface coordinate system (ρ). The radial impurity transport equation for tokamak plasmas is a set of non-linear, parabolic, partial differential equations, solving which generates the radial distributions of all impurity charge states (Z) within the plasma. The present study illustrates the application of a semi-implicit method over the ρ-based impurity transport equation, generated by applying a transformation of the coordinate for the poloidal cross-section of the torus-shaped plasma confinement system, from its geometric radius (r) to the magnetic flux surface coordinate system (ρ). The study further discusses the von Neumann stability analysis of the numerical scheme applied to this transformed (ρ-based) impurity transport equation. The von Neumann stability analysis of the semi-implicit formulation of the radial impurity transport equation has been reported earlier. The stability condition derived in this study is, therefore, a generalization to the earlier reported stability condition now applicable to all ρ(r) including the specific case ρ = r considered in the earlier study. The effects of the impurity transport coefficient (D and v) profiles and the plasma and impurity parameter profiles on the derived ρ-based stability condition are analysed in this study. The impurity element considered is oxygen (1 ≤ Z ≤ 8) and the geometry and plasma parameters of the ADITYA tokamak are applied to the cases studied for consistency. [ABSTRACT FROM AUTHOR]

In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.

In this work, we use two different techniques to discuss approximate analytical solutions for the time‐fractional Fokker–Planck equation (TFFPE), namely the new iterative method (NIM) and the fractional power series method (FPSM). Stability analyses and truncation errors are studied using a procedure like the fundamental von Neumann stability analysis. Discretization is carried out numerically for TFFPE by the implicit finite difference and the Crank–Nicolson method. The techniques used in solving the TFFPE are simple and powerful enough to understand the numerical solutions of linear and nonlinear fractional differential equations. We discuss the approximate solutions obtained using the NIM and FPSM. This is explained by employing tables and shapes. The approximate solutions strongly converge to an accurate solution. All computations in this work were carried out using Maple 16. [ABSTRACT FROM AUTHOR]

We give a proof of the existence of a solution of reconstruction operators used in the PNPM DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the PNPM DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several PNPM DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

• A new stabilization term proportional to h 1 − α , where α is the fractional derivative order, is proposed. • Optimal rates of convergence are theoretically shown for the proposed penalization. • Using a von Neumann analysis, stability conditions for the forward Euler-Local Discontinuous Galerkin method are obtained. • The CFL condition number is studied with respect to the approximation degree and a penalization parameter. • The use of high order strong stability preserving Runge-Kutta methods is illustrated on 1D and 2D problems. A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order 1 < α < 2 is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order h 1 − α and using a von Neumann analysis, stability conditions proportional to h α are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory. [ABSTRACT FROM AUTHOR]

Copyright of Revista Integración is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A mathematical model is considered to determine the effectiveness of disinfectant solution for surface decontamination. The decontamination process involved the diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing effect. The mathematical model is a reaction-diffusion type. Finite difference method and method of lines with fourth-order Runge-Kutta method are utilized to solve the model numerically. To obtain stable solutions, von Neumann stability analysis is employed to evaluate the stability of finite difference method. For stiff problem, Dormand-Prince method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB programming is selected for the computation of numerical solutions. From the results obtained, fourth-order Runge-Kutta method has a larger stability region and better accuracy of solutions compared to finite difference method when solving the disinfectant solution model. Moreover, a numerical simulation is carried out to investigate the effect of different thickness of disinfectant solution on bacteria reduction. Results show that thick disinfectant solution is able to reduce the dimensionless bacteria concentration more effectively. [ABSTRACT FROM AUTHOR]

The radial impurity transport equation for tokamak plasma is a form of diffusion-convection-reaction equation. The impurity transport equation is solved to determine the distribution of impurity (non-fuel) ion species with different ionization states perpendicular to magnetic surfaces of tokamak plasma. The equation for each charge (ionization) state Z is a non-linear, second-order in space, first-order in time, parabolic partial differential equation coupled to the previous Z − 1 and the next Z + 1 charge states of the impurity species through its reaction term. The number of differential equations to be solved simultaneously is hence determined by the number of ionization states of the impurity species studied. The solution to the set of these coupled equations can be obtained using a semi-implicit numerical method applied on it. The present study describes the application of von Neumann stability analysis over the semi-implicit numerical method applied over the radial impurity transport equation and determines a generic stability criterion for the method. The stability analysis is further illustrated using the geometry of Aditya tokamak installed at the Institute for Plasma Research Gandhinagar, India as an example. The impurity species considered is oxygen (Atomic number = 8). This leads to a set of eight coupled equations for charge states Z = 1 to 8 over which von Neumann analysis is illustrated in present study. [ABSTRACT FROM AUTHOR]

A new parametric class of semi-implicit numerical schemes for a level set advection equation on Cartesian grids is derived and analyzed. An accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial Lax–Wendroff procedure and numerical von Neumann stability analysis. The obtained semi-implicit κ -scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one-dimensional scheme that is not the case for analogous fully explicit or fully implicit κ -schemes. A further improvement is obtained by using so-called Corner Transport Upwind extension in two-dimensional case. The extended semi-implicit κ -scheme with a specific (velocity dependent) value of κ is 3rd order accurate in space and time for a constant advection velocity, and it is unconditional stable according to the numerical von Neumann stability analysis for the linear advection equation in general. [ABSTRACT FROM AUTHOR]

Including terrain in atmospheric models gives rise to mesh distortions near the lower boundary that can degrade accuracy and challenge the stability of transport schemes. Multidimensional transport schemes avoid splitting errors on distorted, arbitrary meshes, and method-of-lines schemes have a low computational cost because they perform reconstructions at fixed points. This paper presents a multidimensional method-of-lines finite volume transport scheme, “cubicFit”, which is designed to be numerically stable on arbitrary meshes. Stability conditions derived from a von Neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy in distorted regions of the mesh, and near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage irrespective of the velocity field. The cubicFit scheme is evaluated using three, idealised numerical tests. The first is a variant of a standard horizontal transport test on severely distorted terrain-following meshes. The second is a new test case that assesses accuracy near the ground by transporting a tracer at the lower boundary over steep terrain on terrain-following meshes, cut-cell meshes, and new, slanted-cell meshes that do not suffer from severe time-step constraints associated with cut cells. The third, standard test deforms a tracer in a vortical flow on hexagonal–icosahedral meshes and cubed-sphere meshes. In all tests, cubicFit is stable and largely insensitive to mesh distortions, and cubicFit results are more accurate than those obtained using a multidimensional linear upwind transport scheme. The cubicFit scheme is second-order convergent regardless of mesh distortions. [ABSTRACT FROM AUTHOR]

• A high-order low-dissipation finite-volume scheme for unstructured grids is proposed. • Derivation of adaptive numerical dissipation based on Von Neumann stability analysis. • Efficient control of numerical dissipation via simple neural-network models. • Implementation of the proposed scheme in a 3D flow solver. • Performance benchmarks and comparison to a conventional discretization scheme. Due to their enhanced numerical dissipation properties, high-order discretization methods are an important prerequisite to obtain accurate results with Large-Eddy Simulations. However, the exact amount of dissipation often requires a careful tuning by the user via problem-dependent parameters. In this work we present a fully adaptive dissipation control, which ensures stability and additionally reduces the numerical dissipation to a minimum. This novel approach employs a simple feedforward neural network model, which indirectly tabulates an underlying stability equation and thus reduces the computational overhead to estimate the dissipation during runtime. The methodology is adapted for a high-order k -exact reconstruction method on fully unstructured vertex-centered grids, and it is implemented in a full production flow solver. Based on several test cases, the enhanced accuracy compared to a conventional low-order scheme is demonstrated. Especially when dealing with Large-Eddy Simulation benchmarks, significant savings in computation time and grid resolution requirements can be obtained for reaching a desired level of accuracy. Moreover, compared to a high-order reconstruction method with constant numerical dissipation, the presented adaptive approach consistently yields accurate results, regardless of the flow problem. [ABSTRACT FROM AUTHOR]

The Modified Craig–Sneyd scheme is an alternating direction implicit(ADI) type scheme that was introduced by In ’t Hout and Welfert (2009) [12] in order to numerically solve multidimensional convection–diffusion equations with mixed-derivative terms. It is one of the most prominent ADI schemes currently known for their efficiency in solving above type of problems. This paper deals with a useful stability result for the Modified Craig–Sneyd scheme when applied to two-dimensional convection–diffusion equations with mixed derivative term. The stability of the scheme is analyzed in the von Neumann framework, effectively taking into account the actual size of the mixed derivative term. This study is relevant to an observation of apparent discrepancy in a real world application of the scheme, i.e., in computational finance. The obtained results not only generalize some of the existing stability results, but also clearly justify this surprising observation theoretically. [ABSTRACT FROM AUTHOR]