Numerical Schemes For Partial Differential Equations With Large Time Step And Establishment And Evaluation Of Critical Illness Prediction Model Based On Emergency Medicine Data

In this thesis,we focus on the study of large time step numerical schemes for time-dependent partial differential equations and the prediction of emergency medical data.In the aspect of large time step schemes,we design explicit schemes with large time steps for Vlasov-Poisson equations and nonlinear parabolic equations,respec-tively.At first,we design a new hybrid Hermite essentially weighted non-oscillatory(HWENO)scheme based on the method of lines transpose(MOLT)to approximate one-dimensional linear transport equations and Vlasov-Poisson equations.In the framework of MOLT,we first discretize the time variable implicit,resulting in a boundary value problem at the discrete time level.Then give the explicit integral solution of the bound-ary value problem.Here,we construct the HWENO method in the MOLT framework,which updates the solution and the first-order spatial derivative of the equations at the same time,and applies them to approximate integration.The proposed MOLT-HWENO method has three main advantages.Firstly,even though the scheme can use large time steps of implicit time discretization,we do not need to solve a system on each time level any more.Secondly,the HWENO schemes use a more compact stencil than the WENO schemes with the same order of accuracy.The third one is that the scheme can adapt between the linear scheme and the HWENO scheme automatically,that is the scheme would avoid the oscillations by using HWENO reconstruction nearby discontinuities,and use linear approximation straightforwardly in the smooth regions to increase the ef-ficiency of the scheme.In summary,the MOLT-HWENO scheme has higher efficiency with less numerical errors in smooth regions and less computational costs as well.On the other hand,we propose a class of high order kernel-based explicit unconditionally stable scheme for the nonlinear parabolic equation with variable coefficients.A class of new kernel-based representations is proposed to approximate the spatial derivative,and then combined with the explicit Runge-Kutta time discretization method to approx-imate the nonlinear parabolic equation.Theoretical analysis shows that the method can achieve high order accuracy and unconditional stability by selecting variables appro-priately.As a consequence,compared with other explicit schemes with the same order of accuracy,this method can use large time step to improve the computational effi-ciency.Moreover,without extra computational cost,the proposed scheme can enlarge the available interval of the special parameter in the formulation,leading to less errors and higher efficiency.In the aspect of data analysis,a standardized,systematic and easy data analysis database for emergency medicine is established based on the emergency medical data of the emergency center of the First Affiliated Hospital of University of Science and Technology of China.On the basis of the database,we use multiple logistic regression model to carry out prediction research,and evaluate different models.The results show that the lower bounds of the confidence interval of AUC value of the models are much larger than 0.5,which means that the models have practical value in the statistical sense.At the same time,when the prediction probability of the model is less than 60%,the prediction probability is consistent with the actual probability.