Study On Discontinuous Galerkin Method And Model Reduction Technique For Advection-Diffusion-Reaction Problem

The advection-diffusion-reaction problems are widely used in the fields of fluid mechanics,ecological environment and biological mathematics.At present,domestic and foreign scholars have relatively mature research on advection-diffusion-reaction problems,and there are already various excellent numerical schemes to solve them.The discontinuous Galerkin method is one of the important methods for studying such problems.Overall,the discontinuous Galerkin method has many advantages,such as easier achievement of higher order accuracy,more stable and better ability to capture the discontinuity of the solution.However,the discontinuous Galerkin method also has its own shortcomings,such as its large degree of freedom and high computational cost.Therefore,it is particularly important to design a model reduction method that can maintain the numerical properties of the discontinuous Galerkin method.Based on this background,it is necessary to propose several reduced order discontinuous Galerkin methods of advection-diffusion-reaction problems.The main research content of this dissertation is as follows:Chapter 1 introduces the research background,significance,and current research status of advection-diffusion-reaction problems.At the same time,it also elaborates the research status of discontinuous Galerkin method and model reduction technique.Chapter 2 introduces the local discontinuous Galerkin and Crank-Nicolson methods to construct the fully discrete scheme of the variable coefficient diffusion equation,which is regarded as the original full order model.Subsequently,partial numerical solutions of the original full order model were used as snapshot vectors to get the reduced order basis functions,and the proper orthogonal decomposition method and Galerkin projection were introduced to construct the reduced order model.At the same time,several numerical examples were designed to verify the higher order accuracy of these schemes and the superiority of the reduced order model in computational efficiency.Chapter 3 introduces the hybridizable discontinuous Galerkin and diagonal implicit RungeKutta methods to solve the variable coefficient convection equation.In the original full order model,the elimination method is used to obtain the algebraic equation containing only one unknown quantity,and then the proper orthogonal decomposition method and Galerkin projection are introduced to construct the reduced order model.At the same time,in the numerical test section,multiple sets of initial solutions were given to test the superiority of the proposed numerical schemes in computational performance.Chapter 4 introduces the interior penalty discontinuous Galerkin and the explicit-implicit Runge-Kutta methods to solve the advection-diffusion-reaction equation,obtaining the original full order model with time-space higher order accuracy.Afterwards,the proper orthogonal decomposition method and Galerkin projection will also be introduced to construct the reduced order model,resulting in a fast algorithm that can maintain higher order accuracy.Here,several numerical examples will be designed to verify the effectiveness and superiority of the constructed schemes.Chapter 5 summarizes the main research results of this dissertation and looks at future research.