The effects of the model errors generated by discretization of 'on-off'' processes on VDA

Through an idealized model of a partial differential equation with discontinuous 'on-off'' switches in the forcing term, we investigate the effect of the model error generated by the traditional discretization of discontinuous physical 'on-off'' processes on the variational data assimilation (VDA) in detail. Meanwhile, the validity of the adjoint approach in the VDA with 'on-off'' switches is also examined. The theoretical analyses illustrate that in the analytic case, the gradient of the associated cost function (CF) with respect to an initial condition (IC) exists provided that the IC does not trigger the threshold condition. But in the discrete case, if the on switches (or off switches) in the forward model are straightforwardly assigned the nearest time level after the threshold condition is (or is not) exceeded as the usual treatment, the discrete CF gradients (even the one-sided gradient of CF) with respect to some ICs do not exist due to the model error, which is the difference between the analytic and numerical solutions to the governing equation. Besides, the solution of the corresponding tangent linear model (TLM) obtained by the conventional approach would not be a good first-order linear approximation to the nonlinear perturbation solution of the governing equation. Consequently, the validity of the adjoint approach in VDA with parameterized physical processes could not be guaranteed. Identical twin numerical experiments are conducted to illustrate the influences of these problems on VDA when using adjoint method. The results show that the VDA outcome is quite sensitive to the first guess of the IC, and the minimization processes in the optimization algorithm often fail to converge and poor optimization retrievals would be generated as well. Furthermore, the intermediate interpolation treatment at the switch times of the forward model, which reduces greatly the model error brought by the traditional discretization of 'on-off'' processes, is employed in this study to demonstrate that when the 'on-off'' switches in governing equations are properly numerically treated, the validity of the adjoint approach in VDA with discontinuous physical 'on-off'' processes can still be guaranteed

The effects of the model errors generated by discretization of "on-off'' processes on VDA

International audienceThrough an idealized model of a partial differential equation with discontinuous "on-off'' switches in the forcing term, we investigate the effect of the model error generated by the traditional discretization of discontinuous physical "on-off'' processes on the variational data assimilation (VDA) in detail. Meanwhile, the validity of the adjoint approach in the VDA with "on-off'' switches is also examined. The theoretical analyses illustrate that in the analytic case, the gradient of the associated cost function (CF) with respect to an initial condition (IC) exists provided that the IC does not trigger the threshold condition. But in the discrete case, if the on switches (or off switches) in the forward model are straightforwardly assigned the nearest time level after the threshold condition is (or is not) exceeded as the usual treatment, the discrete CF gradients (even the one-sided gradient of CF) with respect to some ICs do not exist due to the model error, which is the difference between the analytic and numerical solutions to the governing equation. Besides, the solution of the corresponding tangent linear model (TLM) obtained by the conventional approach would not be a good first-order linear approximation to the nonlinear perturbation solution of the governing equation. Consequently, the validity of the adjoint approach in VDA with parameterized physical processes could not be guaranteed. Identical twin numerical experiments are conducted to illustrate the influences of these problems on VDA when using adjoint method. The results show that the VDA outcome is quite sensitive to the first guess of the IC, and the minimization processes in the optimization algorithm often fail to converge and poor optimization retrievals would be generated as well. Furthermore, the intermediate interpolation treatment at the switch times of the forward model, which reduces greatly the model error brought by the traditional discretization of "on-off'' processes, is employed in this study to demonstrate that when the "on-off'' switches in governing equations are properly numerically treated, the validity of the adjoint approach in VDA with discontinuous physical "on-off'' processes can still be guaranteed

Numerical investigation on impacts of leakage sizes and pressures of fluid conveying pipes on aerodynamic behaviors

Small hole leakage of pipes caused by erosion and perforation is the major form leading to the leakage. The leakage rate is an important premise and foundation for consequence computation and risk evaluation. Those published papers fail to systematically study impacts of initial pressures and leakage sizes of a pipe on the leakage rate. More numerical simulation results are not verified by experimental test. This paper applies numerical simulation technology to establish the model of small hole leakage in pipes, designs and processes different leakage modules to simulate different leakage scenes, and then experimentally validates the model correctness. On this basis, this paper studies impacts of initial pressures and leakage sizes on leakage rates and obtains fluid dynamic characteristics around the leakage hole, including velocity distribution and pressure distribution. However, in actual engineering, the position of leakage hole could not be predicted and changed in general. Therefore, this paper further studies impacts of leakage hole positions on the pipe leakage rate. In this way, this research is refined and could provide a theoretical basis for emergency rescue and accident survey of pipe leakage accidents

On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations

AbstractFor Toeplitz system of weakly nonlinear equations, by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew-circulant splitting (CSCS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-CSCS and nonlinear CSCS-like iteration methods, respectively. The advantage of these methods is that they do not require accurate computation and storage of Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Therefore, computational workloads and computer storage may be saved in actual implementations. Theoretical analysis shows that these new iteration methods are local convergent under suitable conditions. Numerical results show that both Picard-CSCS and nonlinear CSCS-like iteration methods are feasible and effective for some cases