A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

CALCOLO ◽  
2021 ◽  
Vol 59 (1) ◽  
Author(s):  
Feng Liao ◽  
Fazhan Geng ◽  
Tingchun Wang
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Compact Difference Scheme ◽  
Dirac Equations ◽  
Compact Difference ◽  
Klein Gordon

Three-dimensional forward modelling of gravity field vector and its gradient tensor using the compact difference schemes

2020 ◽  
Vol 224 (2) ◽  
pp. 1272-1286
Author(s):  
Kejia Pan ◽  
Zhihao Zhang ◽  
Shuanggui Hu ◽  
Zhengyong Ren ◽  
Rongwen Guo ◽  
...  
Keyword(s):  
Difference Scheme ◽  
Gravity Field ◽  
Linear Systems ◽  
Fourth Order ◽  
Field Vector ◽  
Forward Modelling ◽  
Compact Difference Scheme ◽  
Order Accuracy ◽  
Gradient Tensor ◽  
Compact Difference

SUMMARY The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

Sign in / Sign up

Export Citation Format

Share Document