Arbitrary source and receiver positioning in finite‐difference schemes using Kaiser windowed sinc functions

Geophysics ◽  
2002 ◽  
Vol 67 (1) ◽  
pp. 156-165 ◽  
Author(s):  
Graham. J. Hicks
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Source Region ◽  
Seismic Source ◽  
Point Sources ◽  
Finite Difference Schemes ◽  
Surface Source ◽  
Body Forces ◽  
Sinc Functions

In finite‐difference methods a seismic source can be implemented using either initial wavefield values or body forces. However, body forces can only be specified at finite‐difference nodes, and, if using initial values, a source cannot be located close to a reflecting boundary or interface in the model. Hence, difficulties can exist with these schemes when the region surrounding a source is heterogeneous or when a source either is positioned between nodes or is arbitrarily close to a free surface. A completely general solution to these problems can be obtained by using Kaiser windowed sinc functions to define a small region around the true source location that contains several nodal body forces. Both monopole and dipole point sources can be defined, enabling many source types to be implemented in either acoustic or elastic media. Such a function can also be used to arbitrarily locate receivers. If the number of finite‐difference nodes per wavelength is four or more (and with a source region half‐width of only four nodes) this scheme results in insignificant phase errors and in amplitude errors of no more than 0.1%. Numerical examples for sources located less than one node from either a free surface or an image source demonstrate that the scheme can be used successfully for any surface‐source or multisource configuration.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

Finite difference methods for 2D shallow water equations with dispersion

2019 ◽  
Vol 34 (2) ◽  
pp. 105-117 ◽  
Author(s):  
Gayaz S. Khakimzyanov ◽  
Zinaida I. Fedotova ◽  
Oleg I. Gusev ◽  
Nina Yu. Shokina
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Original System ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Elliptic Type ◽  
Two Dimensional ◽  
Nonlinear Hyperbolic System

Abstract Basic properties of some finite difference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of shallow water equations are qualitatively different in the cases the dispersion is taken into account, or not. The difference in the behavior of phase errors in one- and two-dimensional cases is pointed out. Special attention is paid to the numerical algorithm based on the splitting of the original system of equations into a nonlinear hyperbolic system and a scalar linear equation of elliptic type.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

scholarly journals Finite Difference Methods for Weather Prediction in Abuja, Nigeria

2020 ◽  
pp. 915-931
Author(s):  
Jacob Emmanuel ◽  
Ogunfiditimi F.O. ◽  
Victor Alexander Okhuese ◽  
Odeyemi J. K
Keyword(s):  
Finite Difference ◽  
Weather Prediction ◽  
Finite Difference Methods ◽  
Weather Forecast ◽  
Difference Schemes ◽  
Weather Data ◽  
Finite Difference Schemes ◽  
Numerical Weather ◽  
Difference Methods ◽  
The Finite Difference Method

In this research, we have been able to simulate some finite difference schemes to predict weather trends of Abuja Station, Nigeria. By analyzing the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme hence we use it to simulate numerical weather data obtained from Federal Airports Authority of Nigeria (FAAN), Abuja and corresponding numerical weather data obtained by the compatible finite difference schemes, using MATLAB (R2012a) software to obtain future numerical weather trends.

scholarly journals Simple bespoke preservation of two conservation laws

2018 ◽  
Vol 40 (2) ◽  
pp. 1294-1329 ◽  
Author(s):  
Gianluca Frasca-Caccia ◽  
Peter Ellsworth Hydon
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Vries Equation ◽  
Finite Difference Methods ◽  
Nonlinear Heat Equation ◽  
Finite Difference Schemes ◽  
Simplified Approach ◽  
Numerical Tests ◽  
Difference Methods ◽  
Discrete Conservation Laws

Abstract Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.

A Three-Dimensional Version of the Free Surface Capturing Discretization for Staggered Grid Finite Difference Schemes: Implementation into StagYY

2021 ◽  
Author(s):  
Paul Tackley
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Three Dimensional ◽  
Small Time ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Simulation Code ◽  
Free Surface Capturing

<p>In order to treat a free surface in models of lithosphere and mantle dynamics that use a fixed Eulerian grid it is typical to use "sticky air", a layer of low-viscosity, low-density material above the solid surface (e.g. Crameri et al., 2012). This can, however, cause numerical problems, including poor solver convergence due to the huge viscosity jump and small time-steps due to high velocities in the air. Additionally, it is not completely realistic because the assumed viscosity of the air layer is typically similar to that of rock in the asthenosphere so the surface is not stress free.  </p><p>In order to overcome these problems, Duretz et al. (2016) introduced and tested a method for treating the free surface that instead detects and applies special conditions at the free surface. This avoids the huge viscosity jump and having to solve for velocities in the air. They applied it to a two-dimensional staggered grid finite difference / finite volume scheme, a discretization that is in common use for modelling mantle and lithosphere dynamics. Here I document the application of this approach to a three-dimensional spherical staggered grid solver in the mantle simulation code StagYY. Some adjustments had to be made to the two-dimensional scheme documented in Duretz et al. (2016) in order to avoid problems due to undefined velocities for certain boundary topographies. The approach was applied not only to the Stokes solver but also to the temperature solver, including the implementation of a mixed radiative/conductive boundary condition applicable to surface magma oceans/lakes.</p><p><strong>References</strong></p><p>Crameri, F., H. Schmeling, G. J. Golabek, T. Duretz, R. Orendt, S. J. H. Buiter, D. A. May, B. J. P. Kaus, T. V. Gerya, and P. J. Tackley (2012), A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ‘sticky air’ method, Geophysical Journal International,189(1), 38-54, doi:10.1111/j.1365-246X.2012.05388.x.</p><p>Duretz, T., D. A. May, and P. Yamato (2016), A free surface capturing discretization for the staggered grid finite difference scheme, Geophysical Journal International, 204(3), 1518-1530, doi:10.1093/gji/ggv526.</p>

A nonstandard finite difference technique for singular Lane-Emden type equations

2019 ◽  
Vol 36 (5) ◽  
pp. 1566-1578 ◽  
Author(s):  
Michael Chapwanya ◽  
Robert Dozva ◽  
Gift Muchatibaya
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Nonlinear Ordinary Differential Equation ◽  
Careful Consideration ◽  
Finite Difference Schemes ◽  
Content Type ◽  
Emden Equation ◽  
Highly Nonlinear ◽  
Parameter Ranges ◽  
Nonstandard Finite Difference

Purpose This paper aims to design new finite difference schemes for the Lane–Emden type equations. In particular, the authors show that the schemes are stable with respect to the properties of the equation. The authors prove the uniqueness of the schemes and provide numerical simulations to support the findings. Design/methodology/approach The Lane–Emden equation is a well-known highly nonlinear ordinary differential equation in mathematical physics. Exact solutions are known for a few parameter ranges and it is important that any approximation captures the properties of the equation it represent. For this reason, designing schemes requires a careful consideration of these properties. The authors apply the well-known nonstandard finite difference methods. Findings Several interesting results are provided in this work. The authors list these as follows. Two new schemes are designed. Mathematical proofs are provided to show the existence and uniqueness of the solution of the discrete schemes. The authors show that the proposed method can be extended to singularly perturbed equations. Originality/value The value of this work can be measured as follows. It is the first time such schemes have been designed for the kind of equations.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

2016 ◽  
Vol 16 (2) ◽  
pp. 231-243
Author(s):  
Francisco José Gaspar ◽  
Francisco Javier Lisbona ◽  
Piotr P. Matus ◽  
Vo Thi Kim Tuyen
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Numerical Approach ◽  
Parabolic Problems ◽  
Finite Difference Schemes ◽  
Quasilinear Parabolic Problems ◽  
Quasilinear Parabolic

AbstractIn this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.

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