Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

1991 ◽  
Vol 12 (5) ◽  
pp. 443-462 ◽  
Author(s):  
A. Cheer ◽  
M. Saleem
Keyword(s):  
Finite Difference ◽  
Gas Dynamics ◽  
Difference Operators ◽  
Acceleration Of Convergence ◽  
Equations Of Gas Dynamics ◽  
Implicit Finite Difference

3-D implicit finite‐difference migration by multiway splitting

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 554-567 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Finite Difference ◽  
Power Series ◽  
Taylor Expansion ◽  
Optimization Technique ◽  
Optimization Procedure ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Square Root ◽  
Implicit Finite Difference ◽  
Root Operator

We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

Implicit finite-difference simulations of seismic wave propagation

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T57-T67 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integration Method ◽  
Time Integration ◽  
Difference Operators ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Time Integration Method ◽  
Implicit Finite Difference ◽  
Splitting Time

We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

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