Advection-diffusion-reaction differential equations arise from petroleum reservoir simulation, groundwater contaminant remediation, and many other applications. The solutions to these kinds of problems usually have moving sharp fronts and cause serious numerical difficulties. Standard numerical methods produce either excessive non-physical oscillations or extra numerical diffusion which smears the sharp fronts. Therefore, many special numerical techniques have been developed to overcome these difficulties. Among them, Eulerian-Lagrangian localized adjoint method (ELLAM) is prominent. In ELLAM, one can use large time steps while maintaining high accuracy. Moreover, all boundary conditions are naturally incorporated into variational forms and mass is conserved.; The major contributions of this dissertation are in the following two areas: In part I, we combine ELLAM framework with multiresolution analysis to develop CFL-free, explicit schemes for time-dependent advection-reaction equations in multiple space dimensions. The developed wavelet schemes include single level scheme, multilevel scheme, and multilevel scheme with adaptive and mass-conservative compression. In part II, we develop two Eulerian-Lagrangian nonoverlapping domain decomposition schemes for unsteady-state advection-diffusion equations in multidimensional spaces.; Included numerical experiments on these schemes and comparison with some other well-received methods demonstrate the strong potentials of our schemes. Theoretical analyses of the schemes are also presented.
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