Cubic Spline Interpolation With Three Rotation Angles (m Method) —Taking the First Derivative as a Parameter

2021 ◽  
Vol 3 (4) ◽  
pp. 1-21
Author(s):  
Liu Dahai
Keyword(s):  
Spline Interpolation ◽  
Cubic Spline ◽  
Cubic Spline Interpolation ◽  
First Derivative

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

2009 ◽  
Vol 25 (2) ◽  
pp. 151-160 ◽  
Author(s):  
T.-L. Tsai ◽  
J.-Y. Chen
Keyword(s):  
Spline Interpolation ◽  
Cubic Spline ◽  
Higher Order ◽  
Difference Approximation ◽  
Central Difference ◽  
Time Line ◽  
Cubic Spline Interpolation ◽  
Numerical Diffusion ◽  
First Derivative ◽  
Endpoint Constraints

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.

On the Use of Temporal Reachout Technique for Characteristics Method with Time-Line Cubic Spline Interpolation

2011 ◽  
Vol 27 (2) ◽  
pp. N5-N11 ◽  
Author(s):  
T.-L. Tsai ◽  
J.-Y. Chen
Keyword(s):  
Spline Interpolation ◽  
Cubic Spline ◽  
Higher Order ◽  
Difference Approximation ◽  
Central Difference ◽  
Time Line ◽  
Cubic Spline Interpolation ◽  
First Derivative ◽  
Characteristics Method ◽  
Simulation Results

ABSTRACTThe study had indicated that the computational performances of the characteristics method with the time-line cubic spline interpolation are related to the endpoint constraint, especially for large Courant number in which the foot of the characteristic trajectory is located near the endpoint. The first derivative endpoint constraint with higher-order central difference approximation provides better simulation results among various endpoint constraints, but it still induces some degree of numerical error. In this study, by locating the foot of the characteristic trajectory away from the endpoint, the temporal reachout technique is proposed to avoid the effect of endpoint constraint on the time-line cubic spline interpolation. Modeling the transport of a Gaussian concentration distribution in a uniform flow with constant diffusion coefficient and the viscous Burgers equation is used to examine the temporal reachout technique. The outcomes show that the temporal reachout technique yields much better simulation results than the first derivative endpoint constraint with higher-order central difference approximation. The effect of endpoint constraint on the time-line cubic spline interpolation can be greatly diminished by the use of the temporal reachout technique.

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