A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncatio...A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncation error is O（△t^2 ＋ △x^4）.展开更多
A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and t...A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).展开更多
In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gam...In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).展开更多
Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, th...Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.展开更多
In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the...In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).展开更多
A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can e...A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.展开更多
Analyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis.Schemes under discussion included the pointwise-and staggered...Analyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis.Schemes under discussion included the pointwise-and staggered-grid type,and were presented in weighted form using candidate schemes with third-order accuracy and three-point stencil.All of these were commonly used in the construction of difference schemes.Criteria for the dispersion overshoot were presented and their critical states were discussed.Two kinds of instabilities were studied due to inverse dissipation,especially those that occur at lower wave numbers.Criteria for the occurrence were presented and the relationship of the two instabilities was discussed.Comparisons were made between the analytical results and the dispersion/dissipation relations by Fourier transformation of typical schemes.As an example,an application of the criteria was given for the remedy of inverse dissipation in Weirs&Mart´ın’s third-order scheme.展开更多
In this paper, the accuracy of Chang＇s unstructured space-time conservation element and solution element （CE/SE） scheme is analysed for the first time. Based on a redefinition of conservation elements and solution ...In this paper, the accuracy of Chang＇s unstructured space-time conservation element and solution element （CE/SE） scheme is analysed for the first time. Based on a redefinition of conservation elements and solution elements, an improved two-dimensional （2D） unstructured CE/SE scheme with an adjustable parameter β is proposed to accurately capture shock waves. The new scheme can be applied to any type of grid without special treatnmnt. Compared with Chang＇s original parameter a, larger/5 dose not cost extra computational resources. Numerical tests reveal that the new scheme is not only clear in physical concept, compact and highly accurate but also more capable of capturing shock waves than the popular fifth-order accurate weighted essentially non-oscillatory scheme.展开更多
Explicit solution techniques have been widely used in geotechnical engineering for simulating the coupled hydro-mechanical(H-M) interaction of fluid flow and deformation induced by structures built above and under sat...Explicit solution techniques have been widely used in geotechnical engineering for simulating the coupled hydro-mechanical(H-M) interaction of fluid flow and deformation induced by structures built above and under saturated ground, i.e. circular footing and deep tunnel. However, the technique is only conditionally stable and requires small time steps, portending its inefficiency for simulating large-scale H-M problems. To improve its efficiency, the unconditionally stable alternating direction explicit(ADE)scheme could be used to solve the flow problem. The standard ADE scheme, however, is only moderately accurate and is restricted to uniform grids and plane strain flow conditions. This paper aims to remove these drawbacks by developing a novel high-order ADE scheme capable of solving flow problems in nonuniform grids and under axisymmetric conditions. The new scheme is derived by performing a fourthorder finite difference(FD) approximation to the spatial derivatives of the axisymmetric fluid-diffusion equation in a non-uniform grid configuration. The implicit Crank-Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps,giving rise to a new axisymmetric ADE scheme. The pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer code fast Lagrangian analysis of continua(FLAC). This coupling procedure is called the sequentially-explicit coupling technique based on the fourth-order axisymmetric ADE scheme or SEA-4-AXI. Application of SEA-4-AXI for solving axisymmetric consolidation of a circular footing and of advancing tunnel in deep saturated ground shows that SEA-4-AXI reduces computer runtime up to 42%-50% that of FLAC’s basic scheme without numerical instability. In addition, it produces high numerical accuracy of the H-M solutions with average percentage difference of only 0.5%-1.8%.展开更多
The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as th...The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past a circular cylinder and a NACA 002 airfoil to verify the reliability of the present high-order algorithm and simulate gas transport phenomena covering various flow regimes. The computed results are found in good agreement both with the theoretical prediction from continuum to rarefied gas dynamics, the related DSMC solutions, and with the experimental results. The numerical effect of the schemes with the different precision and the different types of Boltzmann collision models on the computational efficiency and computed results is investigated and analyzed. The numerical experience indicates that an approach developing and applying the gas-kinetic high-order algorithm is feasible for directly solving the Boltzmann model equation.展开更多
Abstract Based on the Reynolds-averaged Navier--Stokes （RANS） equations and structured grid technology, the calibration and validation of Y-Reo transition model is preformed with fifth-order weighted compact nonline...Abstract Based on the Reynolds-averaged Navier--Stokes （RANS） equations and structured grid technology, the calibration and validation of Y-Reo transition model is preformed with fifth-order weighted compact nonlinear scheme （WCNS）, and the purpose of the present work is to improve the numerical accuracy for aerodynamic characteristics simulation of low-speed flow with transition model on the basis of high-order numerical method study. Firstly, the empirical correlation functions involved in the Y-Reo transition model are modified and calibrated with experimental data of turbulent flat plates. Then, the grid convergence is studied on NLR-7301 two-element airfoil with the modified empirical correlation. At last, the modified empirical correlation is validated with NLR-7301 two-element airfoil and high-lift trapezoidal wing from transition location, velocity pro- file in boundary layer, surface pressure coefficient and aerodynamic characteristics. The numerical results illustrate that the numerical accuracy of transition length and skin friction behind transition location are improved with modified empirical correlation function, and obviously increases the numerical accuracy of aerodynamic characteristics prediction for typical transport configurations in low-speed range.展开更多
基金NSF of the Education Department of Henan Province(20031100010)
文摘A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncation error is O（△t^2 ＋ △x^4）.
文摘A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).
文摘In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).
基金supported by the National Natural Science Foundation of China (Grant No. U1811464)。
文摘Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.
文摘In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).
基金Project supported by the National Natural Science Foundation of China (Nos. 10172015 and 90205010)
文摘A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.
基金This work was sponsored by the National Science Foundation of China under Grant number 10972023 and 11272037also partially supported by National Basic Research Program of China(No.2009CB724100).The first author was very grateful to Prof.Frank Lu for his efforts on the revision of the manuscript.
文摘Analyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis.Schemes under discussion included the pointwise-and staggered-grid type,and were presented in weighted form using candidate schemes with third-order accuracy and three-point stencil.All of these were commonly used in the construction of difference schemes.Criteria for the dispersion overshoot were presented and their critical states were discussed.Two kinds of instabilities were studied due to inverse dissipation,especially those that occur at lower wave numbers.Criteria for the occurrence were presented and the relationship of the two instabilities was discussed.Comparisons were made between the analytical results and the dispersion/dissipation relations by Fourier transformation of typical schemes.As an example,an application of the criteria was given for the remedy of inverse dissipation in Weirs&Mart´ın’s third-order scheme.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10732010 and 10972010)
文摘In this paper, the accuracy of Chang＇s unstructured space-time conservation element and solution element （CE/SE） scheme is analysed for the first time. Based on a redefinition of conservation elements and solution elements, an improved two-dimensional （2D） unstructured CE/SE scheme with an adjustable parameter β is proposed to accurately capture shock waves. The new scheme can be applied to any type of grid without special treatnmnt. Compared with Chang＇s original parameter a, larger/5 dose not cost extra computational resources. Numerical tests reveal that the new scheme is not only clear in physical concept, compact and highly accurate but also more capable of capturing shock waves than the popular fifth-order accurate weighted essentially non-oscillatory scheme.
基金the support from the University Transportation Center for Underground Transportation Infrastructure at the Colorado School of Mines for partially funding this research under Grant No. 69A3551747118 of the Fixing America's Surface Transportation Act (FAST Act) of U.S. DoT FY2016
文摘Explicit solution techniques have been widely used in geotechnical engineering for simulating the coupled hydro-mechanical(H-M) interaction of fluid flow and deformation induced by structures built above and under saturated ground, i.e. circular footing and deep tunnel. However, the technique is only conditionally stable and requires small time steps, portending its inefficiency for simulating large-scale H-M problems. To improve its efficiency, the unconditionally stable alternating direction explicit(ADE)scheme could be used to solve the flow problem. The standard ADE scheme, however, is only moderately accurate and is restricted to uniform grids and plane strain flow conditions. This paper aims to remove these drawbacks by developing a novel high-order ADE scheme capable of solving flow problems in nonuniform grids and under axisymmetric conditions. The new scheme is derived by performing a fourthorder finite difference(FD) approximation to the spatial derivatives of the axisymmetric fluid-diffusion equation in a non-uniform grid configuration. The implicit Crank-Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps,giving rise to a new axisymmetric ADE scheme. The pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer code fast Lagrangian analysis of continua(FLAC). This coupling procedure is called the sequentially-explicit coupling technique based on the fourth-order axisymmetric ADE scheme or SEA-4-AXI. Application of SEA-4-AXI for solving axisymmetric consolidation of a circular footing and of advancing tunnel in deep saturated ground shows that SEA-4-AXI reduces computer runtime up to 42%-50% that of FLAC’s basic scheme without numerical instability. In addition, it produces high numerical accuracy of the H-M solutions with average percentage difference of only 0.5%-1.8%.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10621062 and 91016027)
文摘The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past a circular cylinder and a NACA 002 airfoil to verify the reliability of the present high-order algorithm and simulate gas transport phenomena covering various flow regimes. The computed results are found in good agreement both with the theoretical prediction from continuum to rarefied gas dynamics, the related DSMC solutions, and with the experimental results. The numerical effect of the schemes with the different precision and the different types of Boltzmann collision models on the computational efficiency and computed results is investigated and analyzed. The numerical experience indicates that an approach developing and applying the gas-kinetic high-order algorithm is feasible for directly solving the Boltzmann model equation.
基金supported by the National Basic Research Program of China(No.2014CB744803)
文摘Abstract Based on the Reynolds-averaged Navier--Stokes （RANS） equations and structured grid technology, the calibration and validation of Y-Reo transition model is preformed with fifth-order weighted compact nonlinear scheme （WCNS）, and the purpose of the present work is to improve the numerical accuracy for aerodynamic characteristics simulation of low-speed flow with transition model on the basis of high-order numerical method study. Firstly, the empirical correlation functions involved in the Y-Reo transition model are modified and calibrated with experimental data of turbulent flat plates. Then, the grid convergence is studied on NLR-7301 two-element airfoil with the modified empirical correlation. At last, the modified empirical correlation is validated with NLR-7301 two-element airfoil and high-lift trapezoidal wing from transition location, velocity pro- file in boundary layer, surface pressure coefficient and aerodynamic characteristics. The numerical results illustrate that the numerical accuracy of transition length and skin friction behind transition location are improved with modified empirical correlation function, and obviously increases the numerical accuracy of aerodynamic characteristics prediction for typical transport configurations in low-speed range.
