A FOURTH ORDER DIFFERENCE SCHEME FOR THE MAXWELL EQUATIONS ON YEE GRID

2008 ◽  
Vol 05 (03) ◽  
pp. 613-642 ◽  
Author(s):  
ALY FATHY ◽  
CHENG WANG ◽  
JOSHUA WILSON ◽  
SONGNAN YANG
Keyword(s):  
Fourth Order ◽  
Maxwell Equations ◽  
Numerical Solutions ◽  
Stability Domain ◽  
Time Step ◽  
Runge Kutta Method ◽  
Accuracy Check ◽  
Grid Points ◽  
The Stability ◽  
Magnetic Vectors

The Maxwell equations are solved by a long-stencil fourth order finite difference method over a Yee grid, in which different physical variables are located at staggered mesh points. A careful treatment of the numerical values near the boundary is introduced, which in turn leads to a "symmetric image" formula at the "ghost" grid points. Such a symmetric formula assures the stability of the boundary extrapolation. In turn, the fourth order discrete curl operator for the electric and magnetic vectors gives a complete set of eigenvalues in the purely imaginary axis. To advance the dynamic equations, the four-stage Runge–Kutta method is utilized, which results in a full fourth order accuracy in both time and space. A stability constraint for the time step is formulated at both the theoretical and numerical levels, using an argument of stability domain. An accuracy check is presented to verify the fourth order precision, using a comparison between exact solution and numerical solutions at a fixed final time. In addition, some numerical simulations of a loss-less rectangular cavity are also carried out and the frequency is measured precisely.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

2014 ◽  
Vol 24 (3) ◽  
pp. 635-646 ◽  
Author(s):  
Deqiong Ding ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulation ◽  
Global Stability ◽  
Difference Equations ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Numerical Solutions ◽  
Time Step ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

Abstract In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results

scholarly journals Numerical Solution Strategies in Permeation Processes

2021 ◽  
Vol 413 ◽  
pp. 29-46
Author(s):  
Axel von der Weth ◽  
Daniela Piccioni Koch ◽  
Frederik Arbeiter ◽  
Till Glage ◽  
Dmitry Klimenko ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Ad Hoc ◽  
Numerical Solutions ◽  
Matrix Multiplication ◽  
Transport Processes ◽  
Time Step ◽  
Stability Range ◽  
The Stability ◽  
The One ◽  
Crank Nicolson

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

scholarly journals A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics

2007 ◽  
Vol 7 (3) ◽  
pp. 227-238 ◽  
Author(s):  
S.H. Razavi ◽  
A. Abolmaali ◽  
M. Ghassemieh
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Time Integration ◽  
Critical Time ◽  
Linear Acceleration ◽  
Time Step ◽  
Time Integration Method ◽  
Acceleration Method ◽  
The Stability ◽  
Critical Time Step

AbstractIn the proposed method, the variation of displacement in each time step is assumed to be a fourth order polynomial in time and its five unknown coefficients are calculated based on: two initial conditions from the previous time step; satisfying the equation of motion at both ends of the time step; and the zero weighted residual within the time step. This method is non-dissipative and its dispersion is considerably less than in other popular methods. The stability of the method shows that the critical time step is more than twice of that for the linear acceleration method and its convergence is of fourth order.

scholarly journals Sin-Cos-Taylor-Like method for solving stiff ordinary diffrential equations

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Rokiah @ Rozita Ahmad ◽  
Nazeeruddin Yaacob
Keyword(s):  
Fourth Order ◽  
Stability Property ◽  
Stiff Problems ◽  
Explicit Method ◽  
Implicit Methods ◽  
Stiff Ordinary Differential Equations ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability ◽  
The Cost

This paper discusses the derivation of an explicit Sin-Cos-Taylor-Like method for solving stiff ordinary differential equations, which is a formulation of the combination of a polynomial and the exponential function. This new method requires extra work to evaluate a number of differentiations of the function involved. However, the result shows smaller errors when compared to the results from the explicit classical fourth-order Runge-Kutta (RK4) and the Adam-Bashforth-Moulton (ABM) methods. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. Although extra work is required, this explicit method has its own advantages. Besides providing excellent results, the cost of computation using this explicit method is much cheaper than the implicit methods. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge-Kutta method is not adequate for the solution of stiff problems. As a result, we find that this explicit method is of order-6, which has been developed, and proved to be both A-stable and L-stable.

scholarly journals Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1039
Author(s):  
Jiawei Liu ◽  
Wen-An Yong ◽  
Jianxin Liu ◽  
Zhenwei Guo
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Energy Method ◽  
Numerical Solutions ◽  
Stability Conditions ◽  
Time Step ◽  
Characteristic Boundary ◽  
The Stability

In this paper, a new stable finite-difference (FD) method for solving elastodynamic equations is presented and applied on the Biot and Biot/squirt (BISQ) models. This method is based on the operator splitting theory and makes use of the characteristic boundary conditions to confirm the overall stability which is demonstrated with the energy method. Through the stability analysis, it is showed that the stability conditions are more generous than that of the traditional algorithms. It allows us to use the larger time step τ in the procedures for the elastic wave field solutions. This context also provides and compares the computational results from the stable Biot and unstable BISQ models. The comparisons show that this FD method can apply a new numerical technique to detect the stability of the seismic wave propagation theories. The rigorous theoretical stability analysis with the energy method is presented and the stable/unstable performance with the numerical solutions is also revealed. The truncation errors and the detailed stability conditions of the FD methods with different characteristic boundary conditions have also been evaluated. Several applications of the constructed FD methods are presented. When the stable FD methods to the elastic wave models are applied, an initial stability test can be established. Further work is still necessary to improve the accuracy of the method.

scholarly journals Dynamic Stability of Plane Free Surface of Liquid in Axisymmetric Tanks

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Siva Srinivas Kolukula ◽  
P. Chellapandi
Keyword(s):  
Free Surface ◽  
Parametric Instability ◽  
Nonlinear Finite Element ◽  
Linear Potential ◽  
Kutta Method ◽  
Nonlinear Finite Element Method ◽  
Time Step ◽  
Runge Kutta Method ◽  
Numerical Instabilities ◽  
The Stability

When liquid filled containers are excited vertically, it is known that, for some combinations of frequency and amplitude, the free surface undergoes unbounded motion leading to instability, called parametric instability or parametric resonance, while for other combinations the free surface remains plane. In this paper, the stability of the plane free surface is investigated theoretically when the vessel is a vertical axisymmetric container. The effect of coupled horizontal excitation on the stability is examined. The dynamics of sloshing flows under specified excitations are simulated numerically using fully nonlinear finite element method based on non-linear potential flow theory. A mixed Eulerian-Lagrangian technique combined with 4th-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is applied to the free surface for every finite time step to avoid possible numerical instabilities.

scholarly journals Analysis of a Novel Two-Lane Hydrodynamic Lattice Model Accounting for Driver’s Aggressive Effect and Flow Difference Integral

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Xinyue Qi ◽  
Hongxia Ge ◽  
Rongjun Cheng
Keyword(s):  
Grid Point ◽  
Mkdv Equation ◽  
Neutral Stability ◽  
Aggressive Driving ◽  
Time Step ◽  
New Model ◽  
Grid Points ◽  
The Stability ◽  
Relative Flow ◽  
Aggressive Effect

In the actual traffic environment, the driver’s aggressive driving behaviors are closely related to the traffic conditions at the next-nearest grid point at next time step. The driver adjusts the acceleration of the driving vehicle by predicting the density of the front grid points. Considering the driver’s aggressive effect and the relative flow difference integral, a novel two-lane lattice hydrodynamic model is presented in this paper. The linear stability method is used to analyze the current stability of the new model, and the neutral stability curve is obtained. The nonlinear analysis of the new model is carried out by using the theory of perturbations, and the mKdV equation describing the density of the blocked area is derived. The theoretical analysis results are verified by numerical simulation. From the analysis results, it can be seen that the driver’s aggressive effect and the relative flow difference integral can improve the stability of traffic flow comprehensively.

k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge‐effect elimination

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2185-2192 ◽  
Author(s):  
B. Compani‐Tabrizi
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Time Step ◽  
Absorbing Boundary ◽  
Temporal Derivative ◽  
Time Domain Scattering ◽  
The Stability ◽  
Spatially Varying ◽  
And Storage

The solution algorithm to the absorptive acoustic scalar wave equation with spatially varying velocity and absorptive fields is numerically examined in the context of the k-space time‐domain scattering formalism to construct an absorbing boundary potential which eliminates wraparound and edge effects. The absorptive potential is constructed by using the absorptive coefficient, i.e., the coefficient of the first temporal derivative in the differential equation. Numerical solutions, in two dimensions, show the stability of the algorithm and the elimination of wraparound and edge reflections through use of the constructed absorptive potential. The numbers of calculations and storage requirements per time step are on the order of [Formula: see text] and N, respectively, where N is the number of points into which the problem is discretized.

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