The conventional nonstationary convolutional model assumes that the seismic signal is recorded at normal incidence. Raw shot gathers are far from this assumption because of the effects of offsets. Because of such prob...The conventional nonstationary convolutional model assumes that the seismic signal is recorded at normal incidence. Raw shot gathers are far from this assumption because of the effects of offsets. Because of such problems, we propose a novel prestack nonstationary deconvolution approach. We introduce the radial trace （RT） transform to the nonstationary deconvolution, we estimate the nonstationary deconvolution factor with hyperbolic smoothing based on variable-step sampling （VSS） in the RT domain, and we obtain the high-resolution prestack nonstationary deconvolution data. The RT transform maps the shot record from the offset and traveltime coordinates to those of apparent velocity and traveltime. The ray paths of the traces in the RT better satisfy the assumptions of the convolutional model. The proposed method combines the advantages of stationary deconvolution and inverse Q filtering, without prior information for Q. The nonstationary deconvolution in the RT domain is more suitable than that in the space-time （XT） domain for prestack data because it is the generalized extension of normal incidence. Tests with synthetic and real data demonstrate that the proposed method is more effective in compensating for large-offset and deep data.展开更多
This paper analyzes the stability of milling with variable pitch cutter and tool runout cases characterized by multiple delays,and proposes a new variable-step numerical integration method for efficient and accurate s...This paper analyzes the stability of milling with variable pitch cutter and tool runout cases characterized by multiple delays,and proposes a new variable-step numerical integration method for efficient and accurate stability prediction. The variable-step technique is emphasized here to expand the numerical integration method,especially for the low radial immersion cases with multiple delays. First,the calculation accuracy of the numerical integration method is discussed and the variable-step algorithm is developed for milling stability prediction for single-delay and multiple-delay cases,respectively. The milling stability with variable pitch cutter is analyzed and the result is compared with those predicted with the frequency domain method and the improved full-discretization method. The influence of the runout effect on the stability boundary is investigated by the presented method. The numerical simulation shows that the cutter runout effect increases the stability boundary,and the increasing stability limit is verified by the milling chatter experimental results in the previous research. The numerical and experiment results verify the validity of the proposed method.展开更多
The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial ...The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.展开更多
This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linea...This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2)norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.展开更多
An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectr...An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectral method is applied for the spatial approximation.The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model,primarily due to the introduction of the fractional Laplacian.This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian.Subsequently,the unique solvability and the modified energy dissipation law are theoretically guaranteed.We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term.Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian.In particular,we demonstrate that the adaptive time-stepping strategy,compared with the uniform time steps,captures the multiple time scale evolutions of the solution in simulations.展开更多
In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope selection.We firs...In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope selection.We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint.Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<3.561.Moreover,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0<r_(k)<3.561.This is known to be the best result in the literature.We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.展开更多
基金financially supported by the National Science and Technology Major Project of China(No.2011ZX05023-005-005)the National Natural Science Foundation of China(No.41274137)
文摘The conventional nonstationary convolutional model assumes that the seismic signal is recorded at normal incidence. Raw shot gathers are far from this assumption because of the effects of offsets. Because of such problems, we propose a novel prestack nonstationary deconvolution approach. We introduce the radial trace （RT） transform to the nonstationary deconvolution, we estimate the nonstationary deconvolution factor with hyperbolic smoothing based on variable-step sampling （VSS） in the RT domain, and we obtain the high-resolution prestack nonstationary deconvolution data. The RT transform maps the shot record from the offset and traveltime coordinates to those of apparent velocity and traveltime. The ray paths of the traces in the RT better satisfy the assumptions of the convolutional model. The proposed method combines the advantages of stationary deconvolution and inverse Q filtering, without prior information for Q. The nonstationary deconvolution in the RT domain is more suitable than that in the space-time （XT） domain for prestack data because it is the generalized extension of normal incidence. Tests with synthetic and real data demonstrate that the proposed method is more effective in compensating for large-offset and deep data.
基金supported by the National Key Basic Research Program (Grant No. 2011CB706804)the National Natural Science Foundation of China (Grant No. 50835004)the Ministry of Science and Technology of China (Grant No. 2010ZX04016-012)
文摘This paper analyzes the stability of milling with variable pitch cutter and tool runout cases characterized by multiple delays,and proposes a new variable-step numerical integration method for efficient and accurate stability prediction. The variable-step technique is emphasized here to expand the numerical integration method,especially for the low radial immersion cases with multiple delays. First,the calculation accuracy of the numerical integration method is discussed and the variable-step algorithm is developed for milling stability prediction for single-delay and multiple-delay cases,respectively. The milling stability with variable pitch cutter is analyzed and the result is compared with those predicted with the frequency domain method and the improved full-discretization method. The influence of the runout effect on the stability boundary is investigated by the presented method. The numerical simulation shows that the cutter runout effect increases the stability boundary,and the increasing stability limit is verified by the milling chatter experimental results in the previous research. The numerical and experiment results verify the validity of the proposed method.
文摘The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.
基金supported by NSF of China under grant number 12071216supported by NNW2018-ZT4A06 project+1 种基金supported by NSF of China under grant numbers 12288201youth innovation promotion association(CAS).
文摘This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2)norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.
基金support by the National Natural Science Foundation of China(Nos.11701081,11861060)the State Key Program of National Natural Science Foundation of China(No.61833005)ZhiShan Youth Scholar Program of SEU.
文摘An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectral method is applied for the spatial approximation.The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model,primarily due to the introduction of the fractional Laplacian.This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian.Subsequently,the unique solvability and the modified energy dissipation law are theoretically guaranteed.We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term.Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian.In particular,we demonstrate that the adaptive time-stepping strategy,compared with the uniform time steps,captures the multiple time scale evolutions of the solution in simulations.
基金supported by National Natural Science Foundation of China(Grant No.12071216)supported by National Natural Science Foundation of China(Grant No.11731006)+2 种基金the NNW2018-ZT4A06 projectsupported by National Natural Science Foundation of China(Grant Nos.11822111,11688101 and 11731006)the Science Challenge Project(Grant No.TZ2018001)。
文摘In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope selection.We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint.Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<3.561.Moreover,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0<r_(k)<3.561.This is known to be the best result in the literature.We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.
