Von Neumann Method Research Articles

Analysis and numerical effects of time-delayed rabies epidemic model with diffusion

Abstract The current work is devoted to investigating the disease dynamics and numerical modeling for the delay diffusion infectious rabies model. To this end, a non-linear diffusive rabies model with delay count is considered. Parameters involved in the model are also described. Equilibrium points of the model are determined and their role in studying the disease dynamics is identified. The basic reproduction number is also studied. Before going towards the numerical technique, the definite existence of the solution is ensured with the help of the Schauder fixed point theorem. A standard result for the uniqueness of the solution is also established. Mapping properties and relative compactness of the operator are studied. The proposed finite difference method is introduced by applying the rules defined by R.E. Mickens. Stability analysis of the proposed method is done by implementing the Von–Neumann method. Taylor’s expansion approach is enforced to examine the consistency of the said method. All the important facts of the proposed numerical device are investigated by presenting the appropriate numerical test example and computer simulations. The effect of τ on infected individuals is also examined, graphically. Moreover, a fruitful conclusion of the study is submitted.

An Unconditionally Stable Conformal LOD-FDTD Method for Curved PEC Objects and its Application to EMC Problems

The traditional finite-difference time-domain (FDTD) method is constrained by the Courant–Friedrich–Levy condition and suffers from the notorious staircase error in electromagnetic simulations. This article proposes a 3-D conformal locally one-dimensional FDTD (CLOD-FDTD) method to address the two issues for modeling perfectly electrical conducting (PEC) objects. By considering the partially filled cells, the proposed CLOD-FDTD method can significantly improve the accuracy compared with the traditional locally one-dimensional FDTD (LOD-FDTD) method and the FDTD method. At the same time, the proposed method preserves unconditional stability, which is analyzed and numerically validated using the von Neumann method. Significant gains in central processing unit time are achieved by using large time steps without sacrificing accuracy. Two numerical examples, including a PEC cylinder and a missile, are used to verify its accuracy and efficiency with different meshes and time steps. It can be found from these examples that the CLOD-FDTD method shows better accuracy and can improve the efficiency compared with those of the traditional FDTD method and the traditional LOD-FDTD method.

A robust Hermite spline collocation technique to study generalized Burgers-Huxley equation, generalized Burgers-Fisher equation and Modified Burgers’ equation

In this paper, a robust Hermite collocation technique is proposed to find the numerical solution of generalized Burgers-Huxley and Burgers-Fisher equations as well as modified Burgers’ equation. In this technique, Hermite collocation method with fifth order Hermite splines have been used to approximate the solution variable and its spatial derivatives. Crank-Nicolson finite difference scheme is applied on time derivatives. The quasilinearization technique is used to linearize the nonlinear terms in the equation. Von-Neumann method is applied to show stability of the proposed technique. Robustness of proposed technique is shown by solving five test examples of these three equations with different parameters. The computed numerical results are better than the results from other techniques compared in this paper and are also matched well with the exact solutions.

A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

This paper presents a new numerical algorithm dealing with multi‐term time‐fractional Burgers‐type equation involving the Caputo derivative. The proposed method consists of temporal discretization of formula and spatial discretization using the exponential B‐splines. The semi implicit approach is applied to discretize the nonlinear term . We adopt the Von–Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method.

An efficient collocation method based on Hermite formula and cubic B-splines for numerical solution of the Burgers’ equation

In this paper, an efficient computational technique for the numerical solution of the Burgers’ equation (BE) is presented. The derivative in space is approximated using cubic B-splines and the Hermite formula whereas time discretization is performed by finite differences. The stability of the proposed scheme is derived using the standard von Neumann method to establish the fact that the errors do not amplify. Convergence analysis for the proposed scheme is also discussed. A sufficiently smooth piecewise continuous function is obtained as an approximation to the exact solution which enables us to approximate the solution at any wanted position in the domain. Numerical tests are carried out to further confirm the accuracy and stability of the method. The outcomes of this study are compared with those previously presented in literature.

Thermal transport controlled by intra- and inter-dot Coulomb interactions in sequential and cotunneling serially-coupled double quantum dots

We study thermoelectric transport through a serial double quantum dot (DQD) coupled to two metallic leads with different thermal energies. We take into account the electron sequential and cotunneling effects via different master equation approaches. In the absence of intra- and inter-dot Coulomb interactions, a small peak in thermoelectric and heat currents is found for EL=ER indicating the Coulomb blockade DQD regime, where EL(ER) is the energy of the state of the left(right) quantum dot. In the presence of intra- and inter-dot Coulomb interactions with strengths Uintra, and Uinter, respectively, avoided crossings or resonance energies between the intra- and the inter-dot two-electron states, 2ES, are found. These resonances induce extra transport channels through the DQD leading to strong side peaks in the thermoelectric and heat currents at EL-ER=±(Uintra-Uinter) in addition to the main peak generated at EL=ER. The current side peaks are enhanced by increased strength of the Coulomb interactions. Interestingly, the current side peaks are enhanced when cotunneling terms are considered in which the resonances of the 2ESs assist the electron cotunneling process through the system. Furthermore, the issue of coherences is carefully checked in the DQD-leads system via different approaches to the master equation, which are the Pauli, the Redfield, a first order Lindblad, and the first- and second order von-Neumann methods. We realize that the Pauli method gives a wrong results for the thermoelectric transport when the role of the coherences is relevant.

A grid based ADI method for the problem of two phase solidification

This paper proposes a transformation based, unconditionally stable, Alternating Direction Implicit (ADI) scheme for solving two-phase Stefan problems of solidification in arbitrary bounded domains. The governing equations of each phase are transformed, from a complex physical domain to a fixed rectangular domain, using body-fitted coordinates. ADI method is used to solve the transformed equations of each phase separately. The unconditional stability of the proposed ADI scheme is discussed numerically using von-Neumann method. Several numerical experiments are carried out for the case of stable solidification to verify the applicability of the proposed method. An excellent agreement has been found between the numerically generated values and the exact/existing solutions. Further, the developed scheme has also been tested on the problems of unstable solidification with mild surface tension and kinetic mobility. Once again the interface location with time has been computed very accurately.

Characteristic-based finite-difference schemes for the simulation of convection–diffusion equation by the finite-difference-based lattice Boltzmann methods

The paper is devoted to the analysis of characteristic-based (CB) schemes for the simulation of the convection–diffusion equation by the lattice Boltzmann method (LBM). Numerical schemes from the first order to fourth one are considered. The stability analysis is realized by the von Neumann method. The stability domains of the schemes are constructed. It is demonstrated that the areas of the stability domains for CB schemes are larger than the domains for the schemes, constructed by the traditional approach, based on the discretization at the Cartesian axes directions. By the solution of the numerical examples with the smooth initial conditions, it is demonstrated that the practical convergence rates of the schemes are consistent with the theoretical values. As it is shown, the proposed schemes can be used for the cases of the Peclet number values, when the classical LBM is unstable.

Two efficient methods for solving the generalized regularized long wave equation

ABSTRACT In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized long wave (GRLW) equation. For this purpose, Bäcklund transformation of the Riccati–Bernoulli equation and sextic B-spline functions are used for the exact and numerical solutions, respectively. The single soliton wave motion is used to confirm the methods which are found to be correct and effective. The three invariants (, and ) of motion have been assessed to indicate the conservation properties of the numerical algorithm. For the motion of single solitary and error norms are handled to evaluate differences between the analytical and numerical solutions. Unconditional stability is demonstrated using von-Neumann method. The procured outcomes show that our new schemes guarantee an evident and a functional mathematical equipment for solving nonlinear evolution equations.

Numerical Stability and Accuracy of CCPR-FDTD for Dispersive Medi

The complex-conjugate pole-residue (CCPR) model has been popularly adopted because CCPR-finite-difference time domain (FDTD) can reduce the memory requirement with the help of complex conjugate property of auxiliary variables. To fully utilize CCPR-FDTD, it is of great necessity to investigate its numerical stability since the FDTD method is conditionally stable. Nonetheless, the numerical stability conditions of CCPR-FDTD have not been studied because its derivation is not straightforward. In this communication, the numerical stability conditions of CCPR-FDTD are systematically derived by combining the von Neumann method with Routh–Hurwitz criterion. It is found that the numerical stability conditions of CCPR-FDTD are the same as those of the modified Lorentz-FDTD with bilinear transform. Moreover, the numerical accuracy of CCPR-FDTD is studied, and numerical examples are employed to validate this work.

Parametric schemes for the simulation of the advection process in finite-difference-based single-relaxation-time lattice Boltzmann methods

The paper is devoted to the analysis of two parametric explicit finite-difference schemes for the linear advection equations, considered on the advection step of the splitting algorithm of the finite-difference-based single-relaxation-time lattice Boltzmann method. The schemes are constructed by the approximation of the terms with the spatial derivatives at the characteristic directions. It is demonstrated that by the proper choice of the parameter values, the third and fourth accuracy orders are realized.The stability analysis is based on the von Neumann method. As a result, the stability conditions as the inequalities on the values of the Courant–Friedrichs–Lewi number are obtained. It is demonstrated that the proposed schemes have better stability properties than the other high-order schemes and schemes with the spatial approximations at the Cartesian axes directions. It is demonstrated that the spurious numerical effects can be diminished by the proper choice of the parameter values.The obtained theoretical results are confirmed by the solution of numerical examples with the smooth and discontinuous initial conditions.

An ADI based body-fitted method for Stefan problem in irregular geometries

This paper deals with a two dimensional Stefan problem of solidification in arbitrary bounded domains. To realize this, the governing equations are transformed from a complex physical domain to a simpler computational domain using body-fitted coordinates and the corresponding transformed equations are solved using an alternating direction implicit (ADI) method. Stability analysis of the proposed ADI scheme, applied to the transformed equations, is discussed using von Neumann method. The developed scheme is then used to solve problems which are initially defined in some regular geometries like rectangular or circular but changes with time to some complex shapes. Comparisons with the existing schemes show that the developed scheme produced better solutions.

Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation

In this paper, we solve modified Burgers equation numerically. Time discretization for modified Burgers equation is made by using finite difference approach along with a linearization technique. For space discretization, we propose two meshless approachs. One of them is based on delta-shaped basis functions-pseudo spectral method and the other is based on barycentric rational interpolation method. To see performance of the proposed methods, two test problems are investigated and obtained results are compared with other studies available in literature such as finite element, wavelet and some collocation methods. Accurate results are obtained using fewer collocation points. Further, von Neumann method has been used to discuss the stability of the methods. The comparisons show the applicability of suggested two methods.

Cubic B-spline Least-square Method Combine with a Quadratic Weight Function for Solving Integro-Differential Equations

In this article, a numerical scheme was implemented for solving the integro-differential equations (IDEs) with the weakly singular kernel by using a new scheme depend on the cubic B-spline least-square method and a quadratic B-spline as a weight function. The numerical results are in suitable agreement with the exact solutions via calculating L2 and L∞ norms errors. Theoretically, we discussed the stability evaluation of the current method using the Von-Neumann method, which explained that this technique is unconditionally stable.

Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines

In this paper, a hybrid numerical technique is developed to solve the Benjamin-Bona-Mahony-Burgers (BBMB) equation. This technique involves the application of quintic Hermite collocation method (QHCM) and weighted finite difference scheme. The discretization of BBMB equation is done using collocation method with Hermite splines of order five along spatial direction and using weighted finite difference scheme along temporal direction. The nonlinear equation is discretized without using the Hopf-Cole transformation. Stability of the proposed technique using Von-Neumann method is checked. Convergence analysis of the technique is also done in detail. Validity of the technique is shown by solving four test examples with different boundary and initial conditions. The calculated results are found to be in good agreement with the exact solutions and are better than the results from other techniques given in the literature.

An efficient numerical technique for solving time fractional Burgers equation

A finite difference scheme which depends on a new approximation based on an extended cubic B-spline for the second order derivative is used to calculate the numerical outcomes of time fractional Burgers equation. The presented scheme uses Caputo’s formulation for the time derivative. Finite difference method will be used to discretize the Caputo’s fractional derivative. The proposed scheme will be shown to be unconditionally stable by Von-Neumann method. The convergence analysis of the numerical scheme will be presented of order O(h2+τ2-α). The presented scheme is tested on four numerical examples. The numerical results are compared favorably with other computational schemes.

Combining Cubic B-Spline Galerkin Method with Quadratic Weight Function for Solving Partial Integro-Differential Equations

In this article, a numerical scheme was implemented for solving the partial integro-differential equations (PIDEs) with weakly singular kernel by using the cubic B-spline Galerkin method with quadratic B-spline as a weight function. backward Euler scheme was used for time direction and the cubic B-spline Galerkin method with quadratic weight function was used for spatial derivative. We observed from the numerical examples that the proposed method possesses a high degree of efficiency and accuracy. In addition, the numerical results are in suitable agreement with the exact solutions via calculating L_2 and〖 L〗_∞ norms errors. Theoretically, we discussed the stable evaluation of the current method using the Von-Neumann method, which explained that the present technique is unconditionally stable.

A fourth-order B-spline collocation method for nonlinear Burgers\u2013Fisher equation

A fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics,gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

Multiple LOD-FDTD Method for Multiconductor Coupled Transmission Lines

This article presents the multiple locally one-dimensional (MLOD) finite-difference time-domain (FDTD) method for multiconductor coupled transmission lines. The method is aptly referred to as the MLOD coupled line (CL)-FDTD method, applicable for general asymmetrical multiconductor lines. The differential equations for multiconductor coupled transmission lines are expressed into general compact matrix form and the appropriate set of split matrices are proposed. Utilizing the proposed split matrices, the update procedures and equations of the MLOD CL-FDTD method are provided for general multiconductor lines. The update procedures comprise number of substeps depending on the number of conductor lines while the left-hand side of the implicit update equation constitutes a tridiagonal matrix that can be solved efficiently. The stability analysis is performed using the von Neumann method in the Fourier domain, as well as the two-media reduced-matrix method in the spatial domain for inhomogeneous media. It is found that our proposed method remains stable beyond the Courant time step. Numerical examples demonstrate that the MLOD CL-FDTD method is able to efficiently simulate time domain fields of multiconductor coupled transmission lines at time step larger than the Courant stability limit while maintaining good accuracy.

Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel

The generalized equal width model is an important non-linear dispersive wave model which is naturally used to describe physical situations in a water channel. In this work, we implement the idea of the interpolation by radial basis function to obtain numerical solution of the non-linear time fractional generalized equal width model defined by Caputo sense. In this technique, firstly, a time discretization is accomplished via the finite difference approach and the non-linear term is linearized by a linearization method. Afterwards, with the help of the radial basis function approximation method is used to discretize the spatial derivative terms. The stability of the method is theoretically discussed using the von Neumann (Fourier series) method. Numerical results and comparisons are presented which illustrate the validity and accuracy of our proposed concepts.