The use of summary representation for electromagnetic modeling

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1506-1516 ◽  
Author(s):  
C. Z. Tarlowski ◽  
A. P. Raiche ◽  
M. Nabighian
Keyword(s):  
Finite Difference ◽  
Computing Time ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
Dimensional Problem ◽  
Hybrid Scheme ◽  
Summary Representation ◽  
Improved Accuracy ◽  
And Storage

The method of summary representation developed by G. N. Polozhii is a quasi‐analytical method for solving self‐adjoint, finite‐difference boundary value problems expressed on regular meshes. In principle, the method should allow considerable savings in computing time as well as improved accuracy when compared to commonly used finite‐difference schemes. We have used summary representation as the basis for a new hybrid scheme to solve the two‐dimensional Helmholtz equation for electromagnetic modeling. The theory behind this hybrid scheme is presented. Preliminary results for the two‐dimensional problem show that substantial computing time and storage savings can be made.

scholarly journals On high-order compact schemes for the multidimensional time-fractional Schrödinger equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rena Eskar ◽  
Xinlong Feng ◽  
Ehmet Kasim
Keyword(s):  
Finite Difference ◽  
High Order ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Step Size ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Fractional Schrödinger Equations ◽  
The One

Abstract In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order $O(\tau ^{2-\alpha }+h^{4})$ O ( τ 2 − α + h 4 ) and $O(\tau ^{3-\alpha }+h^{4})$ O ( τ 3 − α + h 4 ) , respectively, with the temporal step size τ and the spatial step size h, and $\alpha \in (0,1)$ α ∈ ( 0 , 1 ) . For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Hybrid modeling of elastic‐wave propagation in two‐dimensional laterally inhomogeneous media

Geophysics ◽  
1987 ◽  
Vol 52 (6) ◽  
pp. 765-771 ◽  
Author(s):  
B. Kummer ◽  
A. Behle ◽  
F. Dorau
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Hybrid Scheme ◽  
Finite Difference Techniques

We have constructed a hybrid scheme for elastic‐wave propagation in two‐dimensional laterally inhomogeneous media. The algorithm is based on a combination of finite‐difference techniques and the boundary integral equation method. It involves a dedicated application of each of the two methods to specific domains of the model structure; finite‐difference techniques are applied to calculate a set of boundary values (wave field and stress field) in the vicinity of the heterogeneous domain. The continuation of the near‐field response is then calculated by means of the boundary integral equation method. In a numerical example, the hybrid method has been applied to calculate a plane‐wave response for an elastic lens embedded in a homogeneous environment. The example shows that the hybrid scheme enables more efficient modeling, with the same accuracy, than with pure finite‐difference calculations.

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.

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