Fixedpoint fast sweeping WENO methods are a class of efficient highorder numerical methods to solve steadystate solutions of hyperbolic partial differential equations (PDEs). The GaussSeidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steadystate solutions. A nice property of fixedpoint fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when highorder finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse LaxWendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifthorder fixedpoint fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show highorder accuracy and good performance of the method. Furthermore, the method is compared with the popular thirdorder total variation diminishing RungeKutta (TVDRK3) timemarching method for steadystate computations. Numerical examples show that for most of examples, the fixedpoint fast sweeping method saves more than half CPU time costs than TVDRK3 to converge to steadystate solutions.
This paper presents a fast sweeping method (FSM) to calculate the first‐arrival traveltimes of the qP, qSV, and qSH waves in two‐dimensional (2D) transversely isotropic media, whose symmetry axis may have an arbitrary orientation (tilted transverse isotropy [TTI]). The method discretizes the anisotropic eikonal equation with finite difference approximations on a rectangular mesh and solves the discretized system iteratively with the Gauss‐Seidel iterations along alternating sweeping orderings. At each mesh point, a highly nonlinear equation is solved to update the numerical solution until its convergence. For solving the nonlinear equation, an interval that contains the solutions is first determined and partitioned into few subintervals such that each subinterval contains one solution; then, the false position method is applied on these subintervals to compute the solutions; after that, among all possible solutions for the discretized equation, a causality condition is imposed, and the minimum solution satisfying the causality condition is chosen to update the solution. For problems with a point‐source condition, the FSM is extended for solving the anisotropic eikonal equation after a factorization technique is applied to resolve the source singularities, which yields clean first‐order accuracy. When dealing with the triplication of the qSV wave, solutions corresponding to the minimal group velocity are chosen such that continuous solutions are computed. The accuracy, efficiency, and capability of the proposed method are demonstrated with numerical experiments.
more » « less NSFPAR ID:
 10364102
 Publisher / Repository:
 DOI PREFIX: 10.1029
 Date Published:
 Journal Name:
 Journal of Geophysical Research: Solid Earth
 Volume:
 125
 Issue:
 8
 ISSN:
 21699313
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 
In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that a timeharmonic elastic wave equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, motivated by our recent work [Hadamard Babich ansatz for pointsource elastic wave equations in variable media at high frequencies, Multiscale Model Simul. 19/1 (2021) 46–86], we propose a new truncated HadamardBabich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green’s functions of frequencydomain pointsource elastic wave equations in inhomogeneous media in the highfrequency asymptotic regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the HuygensKirchhoff secondarysource principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics can be treated automatically. This yields uniformly accurate solutions both near the source and away from it. The second novelty is that a butterfly algorithm is adapted to accelerate matrixvector products induced by the HuygensKirchhoff integral. The new method enjoys the following desired features: (1) it treats caustics automatically; (2) precomputed asymptotic ingredients can be used to construct Green’s functions of elastic wave equations for many different point sources and for arbitrary frequencies; (3) given a specified number of points per wavelength, it constructs Green’s functions in nearly optimal complexity O(N logN) in terms of the total number of mesh points N, where the prefactor of the complexity depends only on the specified accuracy and is independent of the frequency parameter. Threedimensional numerical examples are presented to demonstrate the performance and accuracy of the new method.more » « less

Firstarrival traveltime tomography is an essential method for obtaining nearsurface velocity models. The adjointstate firstarrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the pointsource isotropic eikonal equation by either ray tracers or eikonal solvers intrinsically corresponds to emanating discrete rays from the source point, the resulting traveltime gradient is singular at the source point, and we denote such a singular pattern the imprint of rayillumination. Because the adjointstate equation propagates traveltime residuals back to the source point according to the negative traveltime gradient, the resulting adjoint state will inherit such an imprint of rayillumination, leading to singular gradientdescent directions when updating the velocity model in the adjointstate traveltime tomography. To mitigate this imprint, we solve the adjointstate equation twice but with different boundary conditions: one being taken to be regular data residuals and the other taken to be ones uniformly, so that we are able to use the latter adjoint state to normalize the regular adjoint state and we further use the normalized quantity to serve as the gradient direction to update the velocity model; we call this process rayillumination compensation. To overcome the issue of limited aperture, we have developed a spatially varying regularization method to stabilize the new gradient direction. A synthetic example demonstrates that our method is able to mitigate the imprint of rayillumination, remove the footprint effect near source points, and provide uniform velocity updates along raypaths. A complex example extracted from the Marmousi2 model and a migration example illustrate that the new method accurately recovers the velocity model and that an offsetdependent inversion strategy can further improve the quality of recovered velocity models.more » « less

This paper develops a treetopological local mesh refinement (TLMR) method on Cartesian grids for the simulation of bioinspired flow with multiple moving objects. The TLMR nests refinement mesh blocks of structured grids to the target regions and arrange the blocks in a tree topology. The method solves the timedependent incompressible flow using a fractionalstep method and discretizes the NavierStokes equation using a finitedifference formulation with an immersed boundary method to resolve the complex boundaries. When iteratively solving the discretized equations across the coarse and fine TLMR blocks, for better accuracy and faster convergence, the momentum equation is solved on all blocks simultaneously, while the Poisson equation is solved recursively from the coarsest block to the finest ones. When the refined blocks of the same block are connected, the parallel Schwarz method is used to iteratively solve both the momentum and Poisson equations. Convergence studies show that the algorithm is secondorder accurate in space for both velocity and pressure, and the developed mesh refinement technique is benchmarked and demonstrated by several canonical flow problems. The TLMR enables a fast solution to an incompressible flow problem with complex boundaries or multiple moving objects. Various bioinspired flows of multiple moving objects show that the solver can save over 80% computational time, proportional to the grid reduction when refinement is applied.more » « less

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely illposed optimization problem (i.e. suffering from possible nonexistence or nonuniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green’s function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an illposed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.more » « less