Second-Order Robust Finite Difference Method for Singularly Perturbed Burger's Equation

2021 ◽  
Author(s):  
Masho Jima Kabeto ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Second Order ◽  
Singularly Perturbed ◽  
Burger's Equation ◽  
Burger’S Equation

Second-generation wavelet optimized finite difference method (SGWOFD) for solution of Burger’s equation with different boundary conditions

2018 ◽  
Vol 16 (04) ◽  
pp. 1850032 ◽  
Author(s):  
Deepika Sharma ◽  
Kavita Goyal
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Test Problem ◽  
Second Generation ◽  
Burger's Equation ◽  
Cpu Time ◽  
Burger’S Equation ◽  
Second Generation Wavelet

In this paper, second-generation wavelet optimized finite difference method (SGWOFD) is developed for solution of Burger’s equation with different boundary conditions. The viscid Burger’s equation is considered with periodic, Dirichlet, Neumann and Robin’s boundary conditions. For the approximations of the differential operators, central finite difference scheme has been used and Crank Nicolson’s scheme is used for the time integration. Numerical solutions have been optimized on an adaptive grid which is generated using the second-generation wavelet. The beauty of the second-generation wavelet is that its construction is not affected by the boundary conditions. For each test problem, the convergence of the method has been verified. The CPU time taken by SGWOFD has been computed for each test problem and is compared with the CPU time taken by the finite difference method on a uniform grid. It has been revealed that SGWOFD is highly efficient.

scholarly journals A comparison of finite-difference and fourier method calculations of synthetic seismograms

1989 ◽  
Vol 79 (4) ◽  
pp. 1210-1230
Author(s):  
C. R. Daudt ◽  
L. W. Braile ◽  
R. L. Nowack ◽  
C. S. Chiang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Group Velocity ◽  
Fourth Order ◽  
Difference Method ◽  
Fourier Method ◽  
Velocity Model ◽  
Second Order ◽  
Heterogeneous Model ◽  
Difference Calculation

Abstract The Fourier method, the second-order finite-difference method, and a fourth-order implicit finite-difference method have been tested using analytical phase and group velocity calculations, homogeneous velocity model calculations for disperson analysis, two-dimensional layered-interface calculations, comparisons with the Cagniard-de Hoop method, and calculations for a laterally heterogeneous model. Group velocity rather than phase velocity dispersion calculations are shown to be a more useful aid in predicting the frequency-dependent travel-time errors resulting from grid dispersion, and in establishing criteria for estimating equivalent accuracy between discrete grid methods. Comparison of the Fourier method with the Cagniard-de Hoop method showed that the Fourier method produced accurate seismic traces for a planar interface model even when a relatively coarse grid calculation was used. Computations using an IBM 3083 showed that Fourier method calculations using fourth-order time derivatives can be performed using as little as one-fourth the CPU time of an equivalent second-order finite-difference calculation. The Fourier method required a factor of 20 less computer storage than the equivalent second-order finite-difference calculation. The fourth-order finite-difference method required two-thirds the CPU time and a factor of 4 less computer storage than the second-order calculation. For comparison purposes, equivalent runs were determined by allowing a group velocity error tolerance of 2.5 per cent numerical dispersion for the maximum seismic frequency in each calculation. The Fourier method was also applied to a laterally heterogeneous model consisting of random velocity variations in the lower half-space. Seismograms for the random velocity model resulted in anticipated variations in amplitude with distance, particularly for refracted phases.

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