Quasi-interpolation operators based on a cubic spline and applications in SAMR simulations

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a b s t r a c t In this paper, we consider the properties of monotonicity-preserving and global conservation preserving for interpolation operators. These two properties play important role when interpolation operators used in many real numerical simulations. In order to attain these two aspects, we propose a one-dimensional (1D) new cubic spline, and extend it to two-dimensional (2D) using tensor-product operation. Based on discrete convolution, 1D and 2D quasi-interpolation operators are presented using these functions. Both analysis and numerical results show that the interpolation operators constructed in this paper are monotonic and conservative. In particular, we consider the numerical simulations of 2D Euler equations based on the technique of structured adaptive mesh refinement (SAMR). In SAMR simulations, effective interpolators are needed for information transportation between the coarser/finer meshes. We applied the 2D quasi-interpolation operator to this environment, and the simulation result show the efficiency and correctness of our interpolator. In many interpolation and approximation problems, some properties, such as the shape-preserving property, of a function should be preserved. That needs to construct special interpolatory functions other than conventional methods. Spline functions play an important role in these problems. There are various spline interpolation or approximation operators which are constructed by convolution [22], widely used in mathematics analysis, statistics and interpolation-approximation theory. Usually, we concern to shape-preserving functions. Research on constructing shape-preserving interpolatory started by Schweikert [21] where exponential splines were used as approximants. There were many subsequent works with exponential and cubic spline interpolants containing ''tension parameters " to control shape, see [15,16,4]. Including shape-preserving spline interpolations, for example, [20] on shape-preserving quadratic spline and [10,17,25,26] on shape preserving C 2 cubic spline interpolation and the references therein. In [1], Andersson and Elfving proposed a method that the interpolation and approximation by monotone cubic splines, where the approximation procedures combine interpolation with least squares approximation. But in these works, they seldom refered to the global conservation-preserving property of interpolation or approximation operators. In many real problems, however, conservation-preserving property—the physical character, is very vital. In the numerical simulations of 0096-3003/$-see front matter Ó 2010 Elsevier Inc. All rights reserved.