We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics...We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.展开更多
Operator-splitting is a popular divide-and-conquer strategy for solving differential equations.Typically,the right-hand side of the differential equation is split into a number of parts that are then integrated separa...Operator-splitting is a popular divide-and-conquer strategy for solving differential equations.Typically,the right-hand side of the differential equation is split into a number of parts that are then integrated separately.Many methods are known that split the right-hand side into two parts.This approach is limiting,however,and there are situations when 3-splitting is more natural and ultimately more advantageous.The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators.It is arguably the most popular method for 3-splitting because of its efficiency,ease of implementation,and intuitive nature.Other 3-splitting methods exist,but they are less well known,and analysis and evaluation of their performance in practice are scarce.We demonstrate the effectiveness of some alternative 3-splitting,second-order methods to Strang splitting on two problems:the reaction-diffusion Brusselator,which can be split into three parts that each have closed-form solutions,and the kinetic Vlasov-Poisson equations that are used in semi-Lagrangian plasma simulations.We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10%–20%over traditional Strang splitting.Our analysis for the practical assessment of the efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.展开更多
A multigrid-assisted solver tbr the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fou...A multigrid-assisted solver tbr the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fourth-order for the viscous terms, and a fractional-step strategy ensures second-order time accuracy. To achieve good time-wise efficiency a multigrid technique is used to solve the highly time-consuming pressure-Poisson equation that requires to be solved at every time step. The speed-up achieved by multigrid is shown in tabular form. The performance and accuracy of the code are first ascertained by computing the flow in a single-sided lid-driven cubic cavity with good grid-economy and comparing the results available in the literature. The code, thus validated, is then applied to a new test problem we propose and various transient and asymptotically obtained steady-state results are presented. Given the care taken to establish the credibility of the code and the good spatio-temporal accuracy of the discretization, these results are accurate and may be used for ascertaining the performance of any computational algorithm applied to this test problem.展开更多
A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L2-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathe...A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L2-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathematically for a nonlinear convection-dominated flow. The error estimation is given and the superiority of this method is verified by numerical test.展开更多
基金Gui-Qiang CHEN was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE(EP/E035027/1)the NSFC under a joint project Grant 10728101+4 种基金the Royal Society-Wolfson Research Merit Award(UK)Changguo XIAO was supported in part by the NSFC under a joint project Grant 10728101Yongqian ZHANG was supported in part by NSFC Project 11031001NSFC Project 11121101the 111 Project B08018(China)
文摘We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.
基金funding from the Natural Sciences and Engineering Research Council of Canada under its Discovery Grant Program(RGPN 2020-04467(RJS)and RGPN 2022-04482(AS))as well as from the US Air Force Office of Scientific Research FA9550-21-1-0031(AS).
文摘Operator-splitting is a popular divide-and-conquer strategy for solving differential equations.Typically,the right-hand side of the differential equation is split into a number of parts that are then integrated separately.Many methods are known that split the right-hand side into two parts.This approach is limiting,however,and there are situations when 3-splitting is more natural and ultimately more advantageous.The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators.It is arguably the most popular method for 3-splitting because of its efficiency,ease of implementation,and intuitive nature.Other 3-splitting methods exist,but they are less well known,and analysis and evaluation of their performance in practice are scarce.We demonstrate the effectiveness of some alternative 3-splitting,second-order methods to Strang splitting on two problems:the reaction-diffusion Brusselator,which can be split into three parts that each have closed-form solutions,and the kinetic Vlasov-Poisson equations that are used in semi-Lagrangian plasma simulations.We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10%–20%over traditional Strang splitting.Our analysis for the practical assessment of the efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.
文摘A multigrid-assisted solver tbr the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fourth-order for the viscous terms, and a fractional-step strategy ensures second-order time accuracy. To achieve good time-wise efficiency a multigrid technique is used to solve the highly time-consuming pressure-Poisson equation that requires to be solved at every time step. The speed-up achieved by multigrid is shown in tabular form. The performance and accuracy of the code are first ascertained by computing the flow in a single-sided lid-driven cubic cavity with good grid-economy and comparing the results available in the literature. The code, thus validated, is then applied to a new test problem we propose and various transient and asymptotically obtained steady-state results are presented. Given the care taken to establish the credibility of the code and the good spatio-temporal accuracy of the discretization, these results are accurate and may be used for ascertaining the performance of any computational algorithm applied to this test problem.
基金the National Natural Science Foundation of China
文摘A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L2-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathematically for a nonlinear convection-dominated flow. The error estimation is given and the superiority of this method is verified by numerical test.
