Numerical Simulation With Adaptive Finite Element Methods For CO₂ Storage Problem And Studies In Slope Limiters Of RKDG Methods

This paper consists of two parts including seven chapters. We are devotedto numerical simulation by adaptive finite element methods for CO₂ storage insaline aquifers and the studies in the slope limiters of RKDG methods.In the first part including Chapters 1-4, we mainly consider numerical simulationof the DDC (dissolution-diffusion-convection) processes in CO₂ storage.In Chapter one, the background and the progress of this problem studies areintroduced. In Chapter two, the physical model and the non-dimensional mathematicalmodel are given, then adaptive finite methods are applied to numericalsimulation of the present mathematical model. Especially, for the algorithm construction,a high resolution static mesh is used on the simulation domain which issensitive to perturbation, while the adaptive mesh is used on the rest. The stabilityand convergence of the given algorithms are shown by some numerical results.In Chapter three, we research the effects of the Rayleigh numbers, the perturbationsgenerated by different random seeds, the aspect ratio and fluctuation onthe considered problem. The relationships between the disturbed intensity andthe onset time of convection, the depth of the diffusive layer and the CO₂ massflux at the top boundary have been found for the first time.In the second part including Chapters 4-6, we mainly study the slope limitersof RKDG methods. Some progresses of computational methods in CFD are reviewedin Chapter four. In Chapter five, the necessary preliminary knowledge inthis part is given, including the introduction of Euler equations and several typicalnumerical examples. The detailed procedures to implement RKDG methods forthe scalar conservation law in one space dimension are given. Finally, the numericalsimulations with RKDG methods for one-dimensional and two-dimensionaltypical examples are realized. In Chapter six, we introduce MBDF limiter, whichis compared with TVD, TVB and BDF limiters by many numerical results. Theh-RKDG methods with TVB and BDF limiters as indicators of adaptive meshare also studied。...