scholarly journals Higher order schemes and Richardson extrapolation for singular perturbation problems

1989 ◽  
Vol 39 (1) ◽  
pp. 129-139 ◽  
Author(s):  
Dragoslav Herceg ◽  
Relja Vulanović ◽  
Nenad Petrović
Keyword(s):  
Finite Difference ◽  
Singular Perturbation ◽  
Fourth Order ◽  
Richardson Extrapolation ◽  
Higher Order ◽  
Difference Schemes ◽  
Singular Perturbation Problems ◽  
Finite Difference Schemes ◽  
Perturbation Problems ◽  
Uniform Accuracy

Semilinear singular perturbation problems are solved numerically by using finite–difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation.

Analysis of higher‐order, finite‐difference schemes in 3-D reverse‐time migration

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 845-856 ◽  
Author(s):  
Wen‐Jing Wu ◽  
Larry R. Lines ◽  
Han‐Xing Lu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Higher Order ◽  
Second Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Scalar Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

The design and usefulness of practical algorithms for 3-D reverse‐time depth migration is examined, while demonstrating data applications of the algorithm. We evaluate quantitatively the accuracy of the finite‐difference operator from second‐order to eighth‐order for the scalar wave equation by comparing numerical and analytical solutions. The results clearly show the advantage of using higher‐order, finite‐difference schemes, especially from second‐order to fourth‐order for space derivatives. Hence, a finite‐difference method with the accuracy of fourth‐order in space and second‐order in time is applied to 3-D full scalar wave equations in reverse‐time migration. Considerable savings in CPU and memory are obtained by using larger horizontal grid spacings than the vertical spacings. Therefore, we derive dispersion and stability conditions for such unequal grid spacing. The 3-D reverse‐time migration of data from the Hibernia Field shows an image improvement over 2-D migrations, despite the fact that much of the structure was considered to be approximately 2-D. Stable, nondispersive, finite‐difference methods of higher order can allow for tractable and efficient 3-D reverse‐time migration solutions.

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