The design of a total energy conserving semi-implicit scheme for the multiple-level baroclinic primitive equation has remained an unsolved problem for a long time. In this work, however, we follow an energy perfect co...The design of a total energy conserving semi-implicit scheme for the multiple-level baroclinic primitive equation has remained an unsolved problem for a long time. In this work, however, we follow an energy perfect conserving semi-implicit scheme of a European Centre for Medium-Range Weather Forecasts (ECMWF) type sigma-coordinate primitive equation which has recently successfully formulated. Some real-data contrast tests between the model of the new conserving scheme and that of the ECMWF-type of global spectral semi-implicit scheme show that the RMS error of the averaged forecast Height at 850 hPa can be clearly improved after the first integral week. The reduction also reaches 50 percent by the 30th day. Further contrast tests demonstrate that the RMS error of the monthly mean height in the middle and lower troposphere also be largely reduced, and some well-known systematical defects can be greatly improved. More detailed analysis reveals that part of the positive contributions comes from improvements of the extra-long wave components. This indicates that a remarkable improvement of the model climate drift level can be achieved by the actual realizing of a conserving time-difference scheme, which thereby eliminates a corresponding computational systematic error source/sink found in the currently-used traditional type of weather and climate system models in relation to the baroclinic primitive equations.展开更多
In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved i...In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved in the computation, and we study the evolution of n-qubit fidelity from the end of one application of the correcting circuit to the end of the next application. We assume that the correcting circuit does not introduce new errors, that it does not increase the execution time (i.e. its application takes zero seconds) and that quantum errors are isotropic. We show that the quantum code increases the fidelity of the states perturbed by quantum errors but that this improvement is not enough to justify the use of quantum codes. Namely, we prove that, taking into account that the time interval between the application of the two corrections is multiplied (at least) by the number of qubits n (due to the coding), the best option is not to use quantum codes, since the fidelity of the uncoded state over a time interval n times smaller is greater than that of the state resulting from the quantum code correction.展开更多
A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors ...A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors in the coupled solutions are higher than an acceptable threshold,then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution.How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution.When L2-norm of the space-time residual over the space-time domain of the computations approaches zero,the approximation φh(x,t)(,)→φ(x,t),the theoretical solution.Thus,the proximity of ||E||L_(2) ,the L_(2)-norm of the space-time residual function,to zero is a measure of the accuracy or the error in the computed solution.In this paper,we present a methodology and a computational framework for computing L2 E in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods.It is shown that the proposed a posteriori computations require h,p,k framework in both space-time coupled as well as space-time decoupled finite element methods to ensure that space-time integrals over space-time discretization are Riemann,hence the proposed a posteriori computations can not be performed in finite difference and finite volume methods of solving initial value problems.High-order global differentiability in time in the integration methods is essential in space-time decoupled method for posterior computations.This restricts the use of methods like Euler’s method,Runge-Kutta methods,etc.,in the time integration of ODE’s in time.Mathematical and computational details including model problem studies are presented in the paper.To authors knowledge,it is the first presentation of the proposed a posteriori error computation methodology and computational infrastructure for initial value problems.展开更多
This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare a...This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare analyzedwith existing methods that are categorized into two different types based on the causes.Then,the new method that bypasses the PGF problem all together is proposed.By comparing these threemethods and analyzing the expression of the scalar gradient in a curvilinear coordinate system,this study finds out that only when using the covariant scalar equations of s-coordinate will the PGF computational form have one term in each momentum component equation,thereby avoiding the PGF problem completely.A convenient way of implementing the covariant scalar equations of s-coordinate in a numerical atmospheric model is illustrated,which is to set corresponding parameters in the scalar equations of the Cartesian coordinate.Finally,two idealized experimentsmanifest that the PGF calculated with the new method is more accurate than using the classic one.This method can be used for oceanic models as well,and needs to be tested in both the atmospheric and oceanic models.展开更多
基金This research was jointly supported by the National Key Programme for Developing Basic Sciences (G1998040911) and the National Natural Science Foundation of China under Grant Nos. 49675267, 49205058, and 49975020.
文摘The design of a total energy conserving semi-implicit scheme for the multiple-level baroclinic primitive equation has remained an unsolved problem for a long time. In this work, however, we follow an energy perfect conserving semi-implicit scheme of a European Centre for Medium-Range Weather Forecasts (ECMWF) type sigma-coordinate primitive equation which has recently successfully formulated. Some real-data contrast tests between the model of the new conserving scheme and that of the ECMWF-type of global spectral semi-implicit scheme show that the RMS error of the averaged forecast Height at 850 hPa can be clearly improved after the first integral week. The reduction also reaches 50 percent by the 30th day. Further contrast tests demonstrate that the RMS error of the monthly mean height in the middle and lower troposphere also be largely reduced, and some well-known systematical defects can be greatly improved. More detailed analysis reveals that part of the positive contributions comes from improvements of the extra-long wave components. This indicates that a remarkable improvement of the model climate drift level can be achieved by the actual realizing of a conserving time-difference scheme, which thereby eliminates a corresponding computational systematic error source/sink found in the currently-used traditional type of weather and climate system models in relation to the baroclinic primitive equations.
文摘In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved in the computation, and we study the evolution of n-qubit fidelity from the end of one application of the correcting circuit to the end of the next application. We assume that the correcting circuit does not introduce new errors, that it does not increase the execution time (i.e. its application takes zero seconds) and that quantum errors are isotropic. We show that the quantum code increases the fidelity of the states perturbed by quantum errors but that this improvement is not enough to justify the use of quantum codes. Namely, we prove that, taking into account that the time interval between the application of the two corrections is multiplied (at least) by the number of qubits n (due to the coding), the best option is not to use quantum codes, since the fidelity of the uncoded state over a time interval n times smaller is greater than that of the state resulting from the quantum code correction.
基金grateful for the facilities provided by the Computational Mechanics Laboratory of the Department of Mechanical Engineering.
文摘A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors in the coupled solutions are higher than an acceptable threshold,then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution.How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution.When L2-norm of the space-time residual over the space-time domain of the computations approaches zero,the approximation φh(x,t)(,)→φ(x,t),the theoretical solution.Thus,the proximity of ||E||L_(2) ,the L_(2)-norm of the space-time residual function,to zero is a measure of the accuracy or the error in the computed solution.In this paper,we present a methodology and a computational framework for computing L2 E in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods.It is shown that the proposed a posteriori computations require h,p,k framework in both space-time coupled as well as space-time decoupled finite element methods to ensure that space-time integrals over space-time discretization are Riemann,hence the proposed a posteriori computations can not be performed in finite difference and finite volume methods of solving initial value problems.High-order global differentiability in time in the integration methods is essential in space-time decoupled method for posterior computations.This restricts the use of methods like Euler’s method,Runge-Kutta methods,etc.,in the time integration of ODE’s in time.Mathematical and computational details including model problem studies are presented in the paper.To authors knowledge,it is the first presentation of the proposed a posteriori error computation methodology and computational infrastructure for initial value problems.
基金supported by the Knowledge Innovation Program of the Chinese Academy of Sciences(KZCX2-YW-Q11-04)the National Basic Research Program of China(973 Program,Grant No.2011CB309704)The second author was supported by the National Natural Science Foundation of China(NSFC)under Grant No.40875022,41175064 and 40633016.
文摘This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare analyzedwith existing methods that are categorized into two different types based on the causes.Then,the new method that bypasses the PGF problem all together is proposed.By comparing these threemethods and analyzing the expression of the scalar gradient in a curvilinear coordinate system,this study finds out that only when using the covariant scalar equations of s-coordinate will the PGF computational form have one term in each momentum component equation,thereby avoiding the PGF problem completely.A convenient way of implementing the covariant scalar equations of s-coordinate in a numerical atmospheric model is illustrated,which is to set corresponding parameters in the scalar equations of the Cartesian coordinate.Finally,two idealized experimentsmanifest that the PGF calculated with the new method is more accurate than using the classic one.This method can be used for oceanic models as well,and needs to be tested in both the atmospheric and oceanic models.
