A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations

Multigrid methods are well known for their high efficiency in solving elliptic boundary value problems. In this study, an improved extrapolation cascadic multigrid (EXCMG) method is presented to solve large sparse systems of linear equations, which are discretized from both 2.5-D and 3-D DC resistivity modelling using the finite element methods. To increase the accuracy, the singularity generated by the source term is removed by reformulating the solution with the secondary potential. In addition, a set of new and efficient Fourier coefficient is presented to transform the solutions in the 2.5-D Fourier domain to the 3-D Cartesian domain. To show the efficiency and the ease-to-implement of EXCMG, we first implement the EXCMG methods to a two-layered model of both 2-D and 3-D and compare the results with the analytical solutions. It has been shown that the maximum relative error in apparent resistivity is no more than 0.4 per cent provided an appropriate grid size is chosen. Then the comparisons of EXCMG with two other iterative solvers [symmetric successive over-relaxation conjugate gradient (SSORCG) and incomplete Cholesky conjugate gradient (ICCG)] show that converging at a rate independent of the grid size, the EXCMG method is much more efficient than SSORCG and ICCG solvers. Moreover, the EXCMG method has been shown its potential for being generalized to large-scale 3-D problems, due to the fact that it becomes more efficient as the size of the problem increases.

On the convergence of an extrapolation cascadic multigrid method for elliptic problems

An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients

Efficient forward modeling of electromagnetic (EM) fields is the basis of interpretation and inversion of the EM data, which plays an important role in practical geophysical exploration. A novel extrapolation cascadic multigrid (EXCMG) method is developed to solve large linear systems encountered in geophysical EM modeling. The original curl-curl equation with arbitrary anisotropic conductivity is regularized by including the gradient of a scaled divergence correction term, and a linear edge element method is used to discretize the equation. We develop a novel approach to address the issue of edge unknowns in 3D edge element discretizations on nonuniform rectilinear grids. Inspired by the original EXCMG method for nodal elements, we introduce a new prolongation operator that treats edge unknowns as defined on the midpoints of edges. This operator aims to provide an accurate approximation of the finite-element solution on the refined grid. Using a good initial guess significantly reduces the number of iterations required by the preconditioned biconjugate gradient stabilized solver, which is used as a smoother for the EXCMG algorithm. Numerical experiments are carried out to validate the accuracy and efficiency of our EXCMG method, including models from magnetotellurics and controlled-source EM modeling. The testing results indicate that EXCMG is more efficient than traditional Krylov-subspace iterative solvers, the algebraic multigrid solver, and those depending on the auxiliary-space Maxwell solver, especially for large-scale problems where the number of unknowns exceeds 10 million. The EXCMG method can also be applied to solve other large-scale forward modeling problems encountered in geophysics.

A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.

We propose a fast and accurate 3D large-scale forward modeling method to compute the gravity gradient tensor, in terms of a fourth-order compact finite difference scheme and the extrapolation cascadic multigrid (EXCMG) method. Firstly, the 19-point fourth-order compact finite difference scheme with unequal mesh sizes in different coordinate directions is used to discretize the governing 3D Poisson equation. The resulted symmetric positive definite linear systems are solved by the EXCMG method. Then, the second-order derivatives (gravity gradient tensor) are calculated by solving a series of tridiagonal linear systems resulting from the fourth-order compact finite difference discretization. Finally, numerical result for a cubic model with positive density is used to verify the accuracy of our presented method. It shows that our method can give nearly fourth-order accurate approximations to gravitational potential and gravity gradient tensor. The accuracy is much higher and more efficient than traditional finite difference methods.

This paper proposes an extrapolation cascadic multigrid (EXCMG) method to solve elliptic problems in domains with reentrant corners. On a class of λ-graded meshes, we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids (current and previous grids). Then, this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method. Recursive application of this idea results in the EXCMG method proposed in this paper. Finally, numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. A new compact scheme, generalizing the compact schemes of Rigal [29], is derived and proved to be unconditionally stable and non-oscillatory. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

Quantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

High order compact finite difference schemes for a system of third order boundary value problem

Two-dimensional incompressible, viscous fluid flow around a rotationally oscillating circular cylinder is studied numerically by using a recently developed higher order compact finite difference scheme (HOC) at Reynolds number Re = 500. The stream function vorticity formulation of Navier Stokes equations in cylindrical polar coordinates are considered as the governing equations. Different values of peak rotation rate αmand forced oscillation frequency feare considered. The simulated results are in a good agreement both qualitatively and quantitatively with the previously published results.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems