The High-end Insurance Monotone Conservative Constant Interpolation Operator

Interpolation plays a very vital role in the theoretical researches and practical applications of function approximation.In the domain of scientific computation, the interpolation operators are usually required to not only satisfy the condition of accuracy, but also possess some properties, such as monotonicity and conservation, the two important aspects, of the functions that they interpolate. The high-order interpolation operators with these properties possess very important applied value when they are used in computational hydrodynamics and numerical forecasts of weather, and so on. The dissertation mainly focuses on this topic—high-order monotonicity- and conservation-preserving interpolation operator, and presents some new methods and techniques.The great contributions of this dissertation are summarized as follows:1. Firstly, we propose a new one-dimensional cubic spline function based on the theories of spline and the properties of convolution. Then we expand it to two dimension by tension product. At last, based on these functions, we obtain the corresponding one-dimensional and two-dimensional quasi-interpolation operators, which are monotonicity-preserving and global conservation-preserving under academic analysis.2. From the view of polynomial interpolation, we present the new one-dimensional interpolation operator and two-dimensional interpolation operator which is extended from the former by in-completed bi-quadratic interpolation. Furthermore, we conclude that the two interpolation operators are monotonicity and conservation-preserving.3. Several one-dimensional monotonicity- and conservation-preserving interpolation operators are derived from some one-dimensional numerical schemes which are used in computational hydrodynamics.4. We take the two-dimensional quasi-interpolation operator and polynomial interpolation operator as refined interpolation operators, used in the software framework of structured adaptive mesh refinement(SAMR), to test the availabilities of the two monotonicity- and conservation-preserving interpolation operators through several concrete numerical examples which are governed by the two-dimensional Euler equation.