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The development of high-order methods (order of accuracy 2nd order) on unstructured grids is widely viewed as a major pacing item in computational fluid dynamics (CFD). Efficient high-order methods capable of handling complex geometries are required to compute vortex dominated flows, and to perform large eddy simulation and direct numerical simulation with complex configurations, and to predict aeroacoustic noise generation and propagation. The primary objective of the present research is to develop implicit, multigrid and adaptive solution algorithms for a promising high-order method, the spectral difference (SD) method. Several major activities were carried out: 1. The development of a hp-adaptation capability to capture both discontinuous and smooth flow features with high efficiency; 2. Development of a accuracy-preserving limiter for discontinuity-capturing for the high-order SD Navier-Stokes solver; 3. Demonstration of the developed code for real world unsteady vortex-dominated flow problems.