Jing-Bo Chen

High-order time discretizations in seismic modeling

Seismic modeling plays an important role in exploration geophysics. High-order modeling schemes are in demand for practical reasons. In this context, I present three kinds of high-order time discretizations: Lax-Wendroff methods, Nystrom methods, and splitting methods. LaxWendroff methods are based on the Taylor expansion and the replacement of high-order temporal derivatives by spatial derivatives, Nystrom methods are simplified Runge-Kutta algorithms, and splitting methods comprise substeps for onestep computation. Based on these methods, three schemes with third-order and fourth-order accuracy in time and pseudospectral discretizations in space are presented. I also compare their accuracy, stability, and computational complexity, and discuss advantages and shortcomings of these algorithms. Numerical experiments show that the fourth-order Lax-Wendroff scheme is more efficient for short-time simulations while the fourth-order Nystrom scheme and the thirdorder splitting scheme are more efficient for long-term computations.

A stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space for the scalar wave equation

Based on the formula for stability of finite-difference methods with second-order in time and general-order in space for the scalar wave equation, I obtain a stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space. Unlike the formula for methods with second-order in time, this formula depends on two parameters: one parameter is related to the weights for approximations of second spatial derivatives; the other parameter is related to the weights for approximations of fourth spatial derivatives. When discretizing the mixed derivatives properly, the formula can be generalized to the case where the spacings in different directions are different. This formula can be useful in high-accuracy seismic modeling using the scalar wave equation on rectangular grids, which involves both high-order spatial discretizations and high-order temporal approximations. I also prove the instability of methods obtained by applying high-order finite-difference approximations directly to the second temporal derivative, and this result solves the “Bording’s conjecture.”

Two kinds of separable approximations for the one-way wave operator

Le Rousseau and de Hoop (2001) developed a generalized screen method that generalizes the phase-screen and the split-step Fourier methods to increase their accuracies with large and rapid lateral variations. Using two Taylor approximations and a perturbation hypothesis, this approach approximates the one-way wave operator by products of functions in space variables and functions in wavenumber variables. This approximation enables the inverse Fourier transform with respect to wavenumbers to be independent of the space variables, thus resulting in significant improvement of the computational efficiency. In spite of its great success, this method has low convergence, and it suffers from the presence of branch points resulting from the choice of the background medium.

Modeling the scalar wave equation with Nyström methods

High-accuracy numerical schemes for modeling of the scalar wave equation based on Nystrom methods are developed in this paper. Space is discretized by using the pseudospectral algorithm. For the time discretization, Nystrom methods are used. A fourth-order symplectic Nystrom method with pseudospectral spatial discretization is presented. This scheme is compared with a commonly used second-order scheme and a fourth-order nonsymplectic Nystrom method. For a typical time-step size, the second-order scheme exhibits spatial dispersion errors for long-time simulations, while both fourth-order schemes do not suffer from these errors. Numerical comparisons show that the fourth-order symplectic algorithm is more accurate than the fourth-order nonsymplectic one. The capability of the symplectic Nystrom method in approximately preserving the discrete energy for long-time simulations is also demonstrated.

A multisymplectic integrator for the periodic nonlinear Schrodinger equation

A multisymplectic integrator for the periodic nonlinear Schrodinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method.

Multisymplectic geometry, local conservation laws and a multisymplectic integrator for the Zakharov-Kuznetsov equation

The multisymplectic geometry for the Zakharov–Kuznetsov equation is presented in this Letter. The multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle. Based on the multisymplectic Hamiltonian formulation, we derive a 36-point multisymplectic integrator.

Total variation in Hamiltonian formalism and symplectic-energy integrators

We present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive symplectic-energy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between the symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplectic-energy integrators is explored.

Square-Conservative Schemes for a Class of Evolution Equations Using Lie-Group Methods

A new method for constructing square-conservative schemes for a class of evolution equations using Lie-group methods is presented. The basic idea is as follows. First, we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere. Second, we apply Lie-group methods to the semidiscrete system, and then square-conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere. Both exponential and Cayley coordinates are used. Numerical experiments are also reported.

A 27-point scheme for a 3D frequency-domain scalar wave equation based on an average-derivative method

Based on an average-derivative method and optimization techniques, a 27-point scheme for a 3D frequency-domain scalar wave equation is developed. Compared to the rotated-coordinate approach, the average-derivative optimal method is not only concise but also applies to equal and unequal directional sampling intervals. The resulting 27-point scheme uses a 27-point operator to approximate spatial derivatives and the mass acceleration term. The coefficients are determined by minimizing phase velocity dispersion errors and the resultant optimal coefficients depend on ratios of directional sampling intervals. Compared to the classical 7-point scheme, the number of grid points per shortest wavelength is reduced from approximately 13 to approximately 4 by this 27-point optimal scheme for equal directional sampling intervals and unequal directional sampling intervals as well. Two numerical examples are presented to demonstrate the theoretical analysis. The average-derivative algorithm is also extended to a 3D frequency-domain viscous scalar wave equation.

Variational integrators and the finite element method

Based on the finite element method, discretizations of Lagrangians in multisymplectic field theory are presented in this paper. In particular, we obtain discrete Lagrangians of high order. The corresponding variational integrators are constructed. The convergence of the variational integrators is discussed. Numerical experiments are also reported.