In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed...In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.展开更多
Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternati...Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point 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 high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point 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 third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.展开更多
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.展开更多
To improve the transmission performance of XCTD channel, this paper proposes a method to measure directly and fit the channel transmission characteristics by using frequency sweeping method. Sinusoidal signals with a ...To improve the transmission performance of XCTD channel, this paper proposes a method to measure directly and fit the channel transmission characteristics by using frequency sweeping method. Sinusoidal signals with a frequency range of 100 Hz to 10 k Hz and an interval of 100 Hz are used to measure transmission characteristics of channels with lengths of 300 m, 800 m, 1300 m, and 1800 m. The correctness of the fitted channel characteristics by transmitting square wave, composite waves of different frequencies, and ASK modulation are verified. The results show that when the frequency of the signal is below 1500 Hz, the channel has very little effect on the signal. The signal compensated for amplitude and phase at the receiver is not as good as the uncompensated signal.Alternatively, when the signal frequency is above 1500 Hz, the channel distorts the signal. The quality of signal compensated for amplitude and phase at receiver is better than that of the uncompensated signal. Thus, we can select the appropriate frequency for XCTD system and the appropriate way to process the received signals. Signals below1500 Hz can be directly used at the receiving end. Signals above 1500 Hz are used after amplitude and phase compensation at the receiving end.展开更多
Fatigue crack propagation characteristics of a diesel engine crankshaft are studied by measuring the fatigue crack growth rate using a frequency sweep method on a resonant fatigue test rig. Based on the phenomenon tha...Fatigue crack propagation characteristics of a diesel engine crankshaft are studied by measuring the fatigue crack growth rate using a frequency sweep method on a resonant fatigue test rig. Based on the phenomenon that the system frequency will change when the crack becomes large, this method can be directly applied to a complex component or structure. Finite element analyses （FEAs） are performed to calibrate the relation between the frequency change and the crack size, and to obtain the natural frequency of the test rig and the stress intensity factor （SIF） of growing cracks. The crack growth rate i.e. da/dN-AK of each crack size is obtained by combining the testing-time monitored data and FEA results. The results show that the crack growth rate of engine crankshaft, which is a component with complex geometry and special surface treatment, is quite different from that of a pure material. There is an apparent turning point in the Paris＇s crack partition. The cause of the fatigue crack growth is also discussed.展开更多
High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping...High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are either fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this paper, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in [18] for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches.展开更多
We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented met...We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.展开更多
We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze t...We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.展开更多
The equilibriummetric forminimizing a continuous congested trafficmodel is the solution of a variational problem involving geodesic distances.The continuous equilibrium metric and its associated variational problem ar...The equilibriummetric forminimizing a continuous congested trafficmodel is the solution of a variational problem involving geodesic distances.The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium.We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation.The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances.The geodesic distance needed for the state equation is computed by solving a factored eikonal equation,and the adjoint state equation is solved by a fast sweeping method.Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.展开更多
In this paper,we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid.The sloshing effect on the free surface of the fluid is neglected.The dynamic behavior of...In this paper,we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid.The sloshing effect on the free surface of the fluid is neglected.The dynamic behavior of the elastic structure is investigated based on the classical shell theory,the constitutive relations of which represent a system of ordinary differential equations written for new unknowns.Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates.Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method.The formulated boundary value problem is solved by Godunov's orthogonal sweep method.Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method.The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions.The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions(simply supported,rigidly clamped,and cantilevered shells)have been studied comprehensively.For conical straight and inverted shells,a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.展开更多
(Re)Ba_(2)Cu_(3)O_(7-x)(REBCO)coated conductors(CCs)have attracted considerable concern because of their outstanding current carrying capacity in magnetic fields of high strengths.A huge electromagnetic force is gener...(Re)Ba_(2)Cu_(3)O_(7-x)(REBCO)coated conductors(CCs)have attracted considerable concern because of their outstanding current carrying capacity in magnetic fields of high strengths.A huge electromagnetic force is generated in the superconducting coil when conducting large currents in strong magnetic field.Thus,management of stress and strain has become a key technical challenge for the stability and safety of superconducting coil during operation.To accurately predict the electro-magnetic and mechanical characteristics of superconducting coil in strong magnetic field,an electromechanical model on the basis of the H-formulation and arbitrary Lagrangian-Eulerian(ALE)method is proposed here with FE software.To verify the proposed model,the simulation outcomes of the coil during magnetization are compared with the experimental outcomes.The coupling effect of magnet at high field strengths is dependent on the position of the coil.To reduce the screening current effect,the overshoot method with plateau is found superior to the traditional overshoot method,and an increase in the stabilization time can decrease the maximum value of stress.Finally,the electromechanical behaviors of single winding coil and two-tapes co-winding coil are compared.展开更多
We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind sche...We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.展开更多
基金supported by the NSF (Grant No.DMS-1753581)supported by NSFC (Grant No.12071392).
文摘In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
基金Research was supported by the NSFC Grant 11872210Research was supported by the NSFC Grant 11872210 and Grant No.MCMS-I-0120G01+1 种基金Research supported in part by the AFOSR Grant FA9550-20-1-0055NSF Grant DMS-2010107.
文摘Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point 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 high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point 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 third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.
文摘Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
基金financially supported by the National Key Research and Development Program of China(Grant No.2016YFC1400400)
文摘To improve the transmission performance of XCTD channel, this paper proposes a method to measure directly and fit the channel transmission characteristics by using frequency sweeping method. Sinusoidal signals with a frequency range of 100 Hz to 10 k Hz and an interval of 100 Hz are used to measure transmission characteristics of channels with lengths of 300 m, 800 m, 1300 m, and 1800 m. The correctness of the fitted channel characteristics by transmitting square wave, composite waves of different frequencies, and ASK modulation are verified. The results show that when the frequency of the signal is below 1500 Hz, the channel has very little effect on the signal. The signal compensated for amplitude and phase at the receiver is not as good as the uncompensated signal.Alternatively, when the signal frequency is above 1500 Hz, the channel distorts the signal. The quality of signal compensated for amplitude and phase at receiver is better than that of the uncompensated signal. Thus, we can select the appropriate frequency for XCTD system and the appropriate way to process the received signals. Signals below1500 Hz can be directly used at the receiving end. Signals above 1500 Hz are used after amplitude and phase compensation at the receiving end.
文摘Fatigue crack propagation characteristics of a diesel engine crankshaft are studied by measuring the fatigue crack growth rate using a frequency sweep method on a resonant fatigue test rig. Based on the phenomenon that the system frequency will change when the crack becomes large, this method can be directly applied to a complex component or structure. Finite element analyses （FEAs） are performed to calibrate the relation between the frequency change and the crack size, and to obtain the natural frequency of the test rig and the stress intensity factor （SIF） of growing cracks. The crack growth rate i.e. da/dN-AK of each crack size is obtained by combining the testing-time monitored data and FEA results. The results show that the crack growth rate of engine crankshaft, which is a component with complex geometry and special surface treatment, is quite different from that of a pure material. There is an apparent turning point in the Paris＇s crack partition. The cause of the fatigue crack growth is also discussed.
文摘High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are either fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this paper, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in [18] for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches.
基金Project (No. MTM2005-08690-C02-02) partially supported by the Spanish Ministry of Science and Innovation Grant
文摘We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.
基金supported by DGICYT MTM2008-03597Ramon y Cajal Programsupported by NSF DMS # 0810104
文摘We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.
基金supported by NSF 0810104 and NSF 1115363Leung was supported in part by Hong Kong RGC under Grant GRF603011HKUST RPC under Grant RPC11SC06.
文摘The equilibriummetric forminimizing a continuous congested trafficmodel is the solution of a variational problem involving geodesic distances.The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium.We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation.The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances.The geodesic distance needed for the state equation is computed by solving a factored eikonal equation,and the adjoint state equation is solved by a fast sweeping method.Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.
基金framework of the government task,registration number of the theme,Grant/Award Number:124020700047-3。
文摘In this paper,we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid.The sloshing effect on the free surface of the fluid is neglected.The dynamic behavior of the elastic structure is investigated based on the classical shell theory,the constitutive relations of which represent a system of ordinary differential equations written for new unknowns.Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates.Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method.The formulated boundary value problem is solved by Godunov's orthogonal sweep method.Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method.The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions.The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions(simply supported,rigidly clamped,and cantilevered shells)have been studied comprehensively.For conical straight and inverted shells,a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.
基金National Natural Science Foundation of China(Nos.U2241267,12172155 and 12302278)National Key Research and Development Program of China(No.2023YFA1607304)+1 种基金Major Scientific and Technological Special Project of Gansu Province(23ZDKA0009)Fundamental Research Funds for the Central Universities(No.lzujbky-2022-48).
文摘(Re)Ba_(2)Cu_(3)O_(7-x)(REBCO)coated conductors(CCs)have attracted considerable concern because of their outstanding current carrying capacity in magnetic fields of high strengths.A huge electromagnetic force is generated in the superconducting coil when conducting large currents in strong magnetic field.Thus,management of stress and strain has become a key technical challenge for the stability and safety of superconducting coil during operation.To accurately predict the electro-magnetic and mechanical characteristics of superconducting coil in strong magnetic field,an electromechanical model on the basis of the H-formulation and arbitrary Lagrangian-Eulerian(ALE)method is proposed here with FE software.To verify the proposed model,the simulation outcomes of the coil during magnetization are compared with the experimental outcomes.The coupling effect of magnet at high field strengths is dependent on the position of the coil.To reduce the screening current effect,the overshoot method with plateau is found superior to the traditional overshoot method,and an increase in the stabilization time can decrease the maximum value of stress.Finally,the electromechanical behaviors of single winding coil and two-tapes co-winding coil are compared.
基金partially supported by ONR Grant N00014-02-1-0090ARO MURI Grant W911NF-07-1-0185NSF Grant DMS0811254
文摘We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.
