scholarly journals Explicit Stencil Computation Schemes Generated by Poisson's Formula for the 2D Wave Equation

2019 ◽  
Author(s):  
Naum Khutoryansky
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integral Representation ◽  
Initial Conditions ◽  
Representation Formula ◽  
Time Step ◽  
Stencil Computation ◽  
Integral Representation Formula ◽  
Time Marching ◽  
First Time

An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented. It is based on using the integral representation formula (Poisson's formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula involving three time levels is also employed. It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid. As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions; 2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme. Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach. Their stability regions are presented. The obtained stencil expressions are compared with the corresponding finite-difference schemes available in the literature. Their novelty features are discussed. Simulation results with new and conventional schemes are presented for two benchmark problems that have exact solutions. It is demonstrated that using the new first time-step calculation procedure instead of the conventional one can provide a significant improvement of accuracy even for later time steps.

scholarly journals Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 364
Author(s):  
Ekaterina Madamlieva ◽  
Mihail Konstantinov ◽  
Marian Milev ◽  
Milena Petkova
Keyword(s):  
Integral Representation ◽  
Distributed Delay ◽  
Representation Formula ◽  
Neutral Systems ◽  
Initial Value ◽  
Fractional Systems ◽  
Integral Representation Formula ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Fractional Neutral Systems

The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives.

Transient Bending Wave Propagation in Anisotropic Plates

1998 ◽  
Vol 65 (4) ◽  
pp. 930-938 ◽  
Author(s):  
K.-E. Fa¨llstro¨m ◽  
O. Lindblom
Keyword(s):  
Experimental Data ◽  
Wave Propagation ◽  
Integral Representation ◽  
Orthotropic Plate ◽  
Representation Formula ◽  
Rotary Inertia ◽  
Anisotropic Plates ◽  
Bending Waves ◽  
Integral Representation Formula ◽  
Bending Wave

In this paper we study transient propagating bending waves. We use the equations of orthotropic plate dynamics, derived by Chow about 25 years ago, where both transverse shear and rotary inertia are included. These equations are extended to include anisotropic plates and an integral representation formula for the bending waves is derived. Chow’s model is compared with the classical Kirchoff’s model. We also investigate the influence of the rotary inertia. Comparisons with experimental data are made as well.

scholarly journals An evaluation of the canadian global meteorological ensemble prediction system for short-term hydrological forecasting

2009 ◽  
Vol 6 (4) ◽  
pp. 4891-4917
Author(s):  
J. A. Velázquez ◽  
T. Petit ◽  
A. Lavoie ◽  
M.-A. Boucher ◽  
R. Turcotte ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Ensemble Prediction ◽  
Prediction System ◽  
Forecast Horizon ◽  
Time Step ◽  
Ensemble Forecasts ◽  
Ensemble Prediction System ◽  
Hydrological Forecasting ◽  
Updating Procedure ◽  
First Time

Abstract. Hydrological forecasting consists in the assessment of future streamflow. Current deterministic forecasts do not give any information concerning the uncertainty, which might be limiting in a decision-making process. Ensemble forecasts are expected to fill this gap. In July 2007, the Meteorological Service of Canada has improved its ensemble prediction system, which has been operational since 1998. It uses the GEM model to generate a 20-member ensemble on a 100 km grid, at mid-latitudes. This improved system is used for the first time for hydrological ensemble predictions. Five watersheds in Quebec (Canada) are studied: Chaudière, Châteauguay, Du Nord, Kénogami and Du Lièvre. An interesting 17-day rainfall event has been selected in October 2007. Forecasts are produced in a 3 h time step for a 3-day forecast horizon. The deterministic forecast is also available and it is compared with the ensemble ones. In order to correct the bias of the ensemble, an updating procedure has been applied to the output data. Results showed that ensemble forecasts are more skilful than the deterministic ones, as measured by the Continuous Ranked Probability Score (CRPS), especially for 72 h forecasts. However, the hydrological ensemble forecasts are under dispersed: a situation that improves with the increasing length of the prediction horizons. We conjecture that this is due in part to the fact that uncertainty in the initial conditions of the hydrological model is not taken into account.

scholarly journals Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

1996 ◽  
Vol 118 (4) ◽  
pp. 622-629 ◽  
Author(s):  
K. J. Baumeister ◽  
K. L. Kreider
Keyword(s):  
Wave Equation ◽  
Steady State ◽  
Finite Difference ◽  
Frequency Domain ◽  
Sound Propagation ◽  
Time Dependent ◽  
Frequency Domain Method ◽  
Impedance Boundary ◽  
Frequency Domain Methods ◽  
Time Marching

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

A Dirichlet Problem for the Inhomogeneous Polyharmonic Equation in the Upper Half Plane

2007 ◽  
Vol 14 (1) ◽  
pp. 33-52
Author(s):  
Heinrich Begehr ◽  
Evgenija Gaertner
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Differential Operator ◽  
Integral Representation ◽  
Half Plane ◽  
Representation Formula ◽  
Higher Order ◽  
Polyharmonic Equation ◽  
Integral Representation Formula ◽  
Upper Half Plane

Abstract On the basis of a higher order integral representation formula related to the polyharmonic differential operator and obtained through a certain polyharmonic Green function, a Dirichlet problem is explicitly solved in the upper half plane.

Axially Symmetric Stress Distributions in Elastic Solids Containing Penny-Shaped Cracks Under Torsion

1975 ◽  
Vol 42 (4) ◽  
pp. 896-897 ◽  
Author(s):  
M. L. Pasha
Keyword(s):  
Stress Intensity ◽  
Integral Representation ◽  
Elastic Medium ◽  
Stress Intensity Factors ◽  
Representation Formula ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Intensity Factors ◽  
Elastic Solids ◽  
Integral Representation Formula

We present the axially symmetric stress distributions in elastic solids containing a pair of axially symmetric penny shaped cracks when the infinite elastic medium is kept under torsion. We derive the integral representation formula for the torsion function and the expressions for the stress-intensity factors.

scholarly journals 2D TESTING OF A FINITE DIFFERENCE SCHEME FOR MODELLING OF A VISCOUS DIFFUSION PROCESS IN COMPRESSIBLE GAS

2020 ◽  
Vol 22 (5) ◽  
pp. 128-131
Author(s):  
V.V. Nikonov ◽  
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Initial Conditions ◽  
Stokes Problem ◽  
Velocity Fields ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Time Step ◽  
Acceptable Accuracy ◽  
Compressible Gas

Viscous subproblem of direct numerical simulation of compressible gas is solved. This subproblem is tested on the two-dimensional problem of impulse start of a flat plate (Stokes’ problem). Three calculations were made with the different initial conditions and velocity fields were obtained. The numerical results are compared with the solution of Stokes’ problem. Analyzing the results, we can conclude that in order to achieve acceptable accuracy, it suffices to choose a time step according to the rule that the author formulated in his earlier works.

scholarly journals The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

2017 ◽  
Author(s):  
Giovanni Di Fratta
Keyword(s):  
Integral Representation ◽  
Explicit Expression ◽  
Representation Formula ◽  
Newtonian Potential ◽  
Bounded Domains ◽  
Dimensional Problem ◽  
Smooth Functions ◽  
Integral Representation Formula ◽  
Transport Theorem ◽  
Stokes Integral

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

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