Adaptive hybridized spline differentiators for numerical solution of the advection equation (Q5893769)

Different kinds of spline differentiators are proposed and developed for the numerical solution of the advection equation (AE), namely, simple polynomial splines, shape-preserving cubic splines, hybridized splines, adaptive grid splines and hybridized splines over an adaptive grid. Different kinds of spline differentiators are used in the numerical solution of the (AE) with three kinds of initial conditions, from which the analytical solution is available for evaluating the accuracy of different numerical methods. Extensive numerical experiments show that the improved shape-preserving spline gives more accurate results than other spline differentiators over a uniform grid. The adaptive grid spline differentiators give much more accurate results than the uniform grid ones within a comparable computing time. The hybridized spline differentiators always reduce the computing time and subside the numerical oscillations. The adaptive grid hybridized splines are the best differentiators, achieving higher accuracy within a shorter computing time.