Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

2005 ◽  
Vol 5 (3) ◽  
pp. 294-330 ◽  
Author(s):  
P. N. Vabishchevich
Keyword(s):  
Differential Operators ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Finite Difference Approximation ◽  
Vector Analysis ◽  
Difference Approximation ◽  
Convection Diffusion ◽  
Irregular Grids ◽  
Grid Operators ◽  
Physics Problems

Abstract Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using gen- eral irregular grids. The vector analysis di®erence operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the di®erence operators constructed on two types of grids are investigated. Construction and analysis of the di®erence schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-diffusion problems. The other examples concerned the solution of the non- stationary vector problems described by the second-order equations or the systems of first-order equations.

scholarly journals Three-dimensional modeling of hydrodynamic problems taking into account elastic processes

2021 ◽  
pp. 1-15
Author(s):  
Yuri Andreevich Poveschenko ◽  
Alexander Yur’evich Krukovskiy ◽  
Dmitri Sergeevich Boykov ◽  
Victoria Olegovna Podryga ◽  
Parvin Ilgar gizi Rahimly
Keyword(s):  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Difference Schemes ◽  
Finite Difference Approximation ◽  
Vector Analysis ◽  
Difference Approximation ◽  
Sound Waves ◽  
Displacement Vectors ◽  
The Difference ◽  
Elastic Processes

A finite-difference approximation of elastic forces on spaced Lagrangian grids is constructed, based on the method of support operators. For displacement vectors on irregular grids, on the topological and geometric structure of which minimal reasonable restrictions are imposed, the approximations of vector analysis operations are constructed in relation to difference schemes for problems of elasticity theory. Taking into account the energy balance of the medium, the constructed families of integrally consistent approximations of vector analysis operations are sufficient for discrete modeling of these processes. The schemes are considered, both using the stress tensor in an explicit form, and dividing it into spherical and shear components (pressure and deviator). The latter is used to construct homogeneous algorithms applicable to both the solid and the vaporized phase. The linear theory of elasticity is used for constructing approximations. The resulting forces in spatial geometry are obtained explicitly. Calculations of the sound waves propagation in a three-dimensional orthogonal aluminum plate due to end impact are presented. These calculations confirm the good quality of the difference schemes constructed in work.

Wavelet-optimized compact finite difference method for convection–diffusion equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mani Mehra ◽  
Kuldip Singh Patel ◽  
Ankita Shukla
Keyword(s):  
Finite Difference ◽  
Two Dimensions ◽  
Finite Difference Approximation ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Uniform Grid ◽  
Adaptive Grids ◽  
Smooth Functions ◽  
Compact Finite Difference ◽  
Convection Diffusion

AbstractIn this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

scholarly journals VAGO METHOD FOR THE SOLUTION OF ELLIPTIC SECOND‐ORDER BOUNDARY VALUE PROBLEMS

2010 ◽  
Vol 15 (4) ◽  
pp. 533-545
Author(s):  
Nikolay Vabishchevich ◽  
Petr Vabishchevich
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Symmetric Tensor ◽  
Second Order ◽  
Diffusion Reaction ◽  
Vector Analysis ◽  
Difference Operators ◽  
Curl Operators ◽  
Grid Operators

Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection‐diffusion‐reaction equation the diffusion coefficient is a symmetric tensor of second order.

scholarly journals Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 485 ◽  
Author(s):  
Eyaya Fekadie Anley ◽  
Zhoushun Zheng
Keyword(s):  
Finite Difference ◽  
Variable Coefficients ◽  
Convective Transport ◽  
Transport Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Convection Diffusion ◽  
Integer Order ◽  
Difference Formula ◽  
The Right

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

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