Investigation of Effect of Endpoint Constraint on Time-Line Cubic Spline Interpolation

Abstract

AbstractIn this study, the effects of various endpoint constraints, including the first derivative, natural, quadratic, and not-a-knot endpoint constraints, on the time-line cubic spline interpolation are examined by solving the advection-diffusion equation with constant and variable velocities, and the viscous Burgers equation. The natural endpoint constraint could produce large numerical diffusion. The quadratic endpoint constraint can decrease the numerical diffusion from the natural endpoint constraint, but it could trigger large numerical oscillation. The first derivative endpoint constraint with higher-order central difference approximation has better simulated results than the natural and quadratic endpoint constraints. It can significantly reduce not only the numerical diffusion produced by the natural endpoint constraint, but also the numerical oscillation caused by the quadratic endpoint constraint. The applicability of the time-line cubic spline interpolation together with the not-a-knot endpoint constraint is limited, since the computational instability is caused while the Courant number is greater than unity. Therefore, as far as accuracy and applicability are concerned, the first derivative endpoint constraint with higher-order central difference approximation rather than the commonly used natural endpoint constraint should be a better choice for the time-line cubic spline interpolation.