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.

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 Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T33-T43
Author(s):  
Chao Lyu ◽  
Yann Capdeville ◽  
Liang Zhao
Keyword(s):  
Wave Equation ◽  
Spectral Element Method ◽  
Computing Time ◽  
Spectral Element ◽  
Two Dimensions ◽  
Elastic Model ◽  
Time Step ◽  
High Degree ◽  
Very High ◽  
Element Method

The spectral element method (SEM) has gained tremendous popularity within the seismological community to solve the wave equation at all scales. Classic SEM applications mostly rely on degrees 4–8 elements in each tensorial direction. Higher degrees are usually not considered due to two main reasons. First, high degrees imply large elements, which make the meshing of mechanical discontinuities difficult. Second, the SEM’s collocation points cluster toward the edge of the elements with the degree, degrading the time-marching stability criteria and imposing a small time step and a high numerical cost. Recently, the homogenization method has been introduced in seismology. This method can be seen as a preprocessing step before solving the wave equation that smooths out the internal mechanical discontinuities of the elastic model. It releases the meshing constraint and makes use of very high degree elements more attractive. Thus, we address the question of memory and computing time efficiency of very high degree elements in SEM, up to degree 40. Numerical analyses reveal that, for a fixed accuracy, very high degree elements require less computer memory than low-degree elements. With minimum sampling points per minimum wavelength of 2.5, the memory needed for a degree 20 is about a quarter that of the one necessary for a degree 4 in two dimensions and about one-eighth in three dimensions. Moreover, for the SEM codes tested in this work, the computation time with degrees 12–24 can be up to twice faster than the classic degree 4. This makes SEM with very high degrees attractive and competitive for solving the wave equation in many situations.

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].

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.

STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS

2013 ◽  
Vol 21 (03) ◽  
pp. 1350003 ◽  
Author(s):  
DANIEL RABINOVICH ◽  
DAN GIVOLI ◽  
THOMAS HAGSTROM ◽  
JACOBO BIELAK
Keyword(s):  
Boundary Condition ◽  
Elastic Waves ◽  
Two Dimensions ◽  
High Order ◽  
Radiation Boundary Condition ◽  
Absorbing Boundary ◽  
Radiation Boundary Conditions ◽  
Long Time ◽  
Radiation Boundary ◽  
The Stability

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

scholarly journals On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Maryam Sarboland ◽  
Azim Aminataei
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Meshfree Methods ◽  
Radial Basis Function Networks ◽  
One Dimensional ◽  
Temporal Derivative ◽  
Spatial Derivatives ◽  
The Stability ◽  
The One ◽  
Nonlinear Nature

The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers’ equation.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

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.

The importance of interfacial instability for viscous folding in mechanically heterogeneous layers

2018 ◽  
Vol 487 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Evangelos Moulas ◽  
Stefan M. Schmalholz
Keyword(s):  
Numerical Solutions ◽  
Pure Shear ◽  
High Amplitude ◽  
Two Dimensions ◽  
Interfacial Instability ◽  
Wavelength Selection ◽  
Interfacial Instabilities ◽  
Time Step ◽  
Heterogeneous Rock ◽  
A Minor

AbstractViscous folding in mechanically heterogeneous layers is modelled numerically in two dimensions for linear and power-law viscous fluids. Viscosity heterogeneities are expressed as circular-shaped variations of the effective viscosity inside and outside the layers. The layers are initially perfectly flat and are shortened in the layer-parallel direction. The viscosity heterogeneities cause a perturbation of the velocity field from the applied bulk pure shear, which perturb geometrically the initially flat-layer interfaces from the first numerical time step. This geometrical perturbation triggers interfacial instabilities, resulting in high-amplitude folding. We compare simulations with heterogeneities with corresponding simulations in which the heterogeneities are removed after the first time step, and, hence, only the initial small geometrical perturbations control wavelength selection and high-amplitude folding. Results for folding in heterogeneous and homogeneous layers are similar, showing that viscosity heterogeneities have a minor to moderate impact on fold wavelength selection and high-amplitude folding. Our results indicate that the interfacial instability is the controlling process for the generation of buckle folds in heterogeneous rock layers. Therefore, existing analytical and numerical solutions for folding in homogeneous layers, in which folding was triggered by geometrical perturbations, are useful and applicable to study folding in natural, heterogeneous rock layers.

Sign in / Sign up

Export Citation Format

Share Document