  A system of high-order adaptive multiresolution wavelet collocation upwind
schemes are developed for the solution of hyperbolic conservation laws. A
couple of asymmetrical wavelet bases with interpolation property are built to
realize the upwind property, and address the nonlinearity in the hyperbolic
problems. An adaptive algorithm based on multiresolution analysis in wavelet
theory is designed to capture moving shock waves and distinguish new localized
steep regions. An integration average reconstruction method is proposed based
on the Lebesgue differentiation theorem to suppress the Gibbs phenomenon. All
these numerical techniques enable the wavelet collocation upwind scheme to
provide a general framework for devising satisfactory adaptive wavelet upwind
methods with high-order accuracy. Several benchmark tests for 1D hyperbolic
problems are carried out to verify the accuracy and efficiency of the present
wavelet schemes.
