In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green fun...In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.展开更多
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency ...A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.展开更多
In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too m...In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too much stability. We will also introduce two more changes, called ENBDF and EBNDF methods. In the first one, the NDF method is used as the first predictor and the BDF as the second predictor. In the EBNDF, the BDF is the first predictor and the NDF is the second one. In both modifications the local truncation error is smaller than in the EBDF. Moreover, the EBNDF method has a larger stability region than the EBDF.展开更多
The relationship between thermal conductivity and properties of mixing particles is required for quantitative study of heat transfer processes in asphalt-based materials. In this paper, we measured the e?ective ther-...The relationship between thermal conductivity and properties of mixing particles is required for quantitative study of heat transfer processes in asphalt-based materials. In this paper, we measured the e?ective ther- mal conductivity of asphalt-based materials with thermal conduction (graphite) and insulation (cenosphere) powders modification. By taking account of the particle shape, volume fraction, the thermal conductivity of filling particles and base asphalt, we present a new differential effective medium formula to predict the thermal conductivity modification in asphalt-based composite. Our theoretical predications are in good agreement with the experiment data. The new model can be applied for predicting the thermal properties of asphalt-based mixture, which is available for most of thermal modification in two-phase composites.展开更多
In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discret...In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.展开更多
文摘In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.
文摘A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.
文摘In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too much stability. We will also introduce two more changes, called ENBDF and EBNDF methods. In the first one, the NDF method is used as the first predictor and the BDF as the second predictor. In the EBNDF, the BDF is the first predictor and the NDF is the second one. In both modifications the local truncation error is smaller than in the EBDF. Moreover, the EBNDF method has a larger stability region than the EBDF.
基金supported by the National Natural Science Foundation of China under grants Nos. 50906073 and 50973018
文摘The relationship between thermal conductivity and properties of mixing particles is required for quantitative study of heat transfer processes in asphalt-based materials. In this paper, we measured the e?ective ther- mal conductivity of asphalt-based materials with thermal conduction (graphite) and insulation (cenosphere) powders modification. By taking account of the particle shape, volume fraction, the thermal conductivity of filling particles and base asphalt, we present a new differential effective medium formula to predict the thermal conductivity modification in asphalt-based composite. Our theoretical predications are in good agreement with the experiment data. The new model can be applied for predicting the thermal properties of asphalt-based mixture, which is available for most of thermal modification in two-phase composites.
基金supported in part by the Computational Physics Key Laboratory of IAPCAM(P.R.China)under Grant 6142A05200103(K.Cheng)the National Science Foundation(USA)under Grant NSF DMS-2012669(C.Wang)Grants NSF DMS-1719854,DMS-2012634(S.Wise).
文摘In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.