We provide an overview of how fluid-solid and fluid-fluid interfaces can be computed successfully with the CIP method based on adaptive Soroban grids. In this approach, the CCUP technique, which is based on the CIP method, is combined with the adaptive Soroban grid technique. One of the superior features of the approach is that even though the grid system is unstructured, it still has a simple data structure that renders remarkable computational efficiency. Another superior feature is that despite the unstructured and collocated nature of the grid, high-order accuracy and computational robustness are maintained. In addition, because the Soroban grid technique does not have any elements or cells connecting the grid points, the approach does not involve mesh distortion limitations. Our objective in this paper is to provide an easy-to-follow description of the key aspects of the approach.

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This paper presents a class of finite volume methods of high accuracy based on the CIP/Multi-Moment concept. The resulting numerical formulations work well for both structured and unstructured meshes and appear to be attractive in computational efficieny and robustness. Numerical tests will be reported as well.

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The IDO (Interpolated Differential Operator) is an Euler-type scheme using local interpolation function constructed by multi moments. A conservative form of the IDO scheme has been developed, in which the point values and the integrated values are employed as dependent variables. The formulation and computational procedure are described. According to the Fourier analysis, it is found that the accuracy and stability are exactly same as that of the conventional non-conservative IDO scheme for linear partial differential equations. The conservative IDO scheme is applied to compressible flow and incompressible flow and better results are obtained.

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In this paper, we present a basis set approach by the Constrained Interpolation Profile (CIP) method to establish a systematic and simple method to get highly accurate numerical solutions of linear and nonlinear partial differential equations. This method uses a simple polynomial basis set, by which physical quantities are approximated with their values and derivatives associated with grid points. Nonlinear operations on functions are carried out in the framework of differential algebra. Then, introducing scalar products and requiring the residue to be orthogonal to the basis, the linear and nonlinear partial differential equations are reduced to ordinary differential equations for values and spatial derivatives. The method is tested on the linear and nonlinear Schrodinger equation and is proven to give highly accurate solutions.

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With the progress of human activities in space, the occasion to handle liquids under low-gravity conditions is now growing. The absence or diminution of gravity force makes it difficult to position and control the liquid in a desirable manner. For the establishment of the technology for the management of liquid propellant in space vehicles, a numerical method, called 'CIP-LSM' (CIP based Level Set method & MARS), was developed to simulate three-dimensional free-surface flows under various gravity conditions, which has been applied to clarify the dynamic byhavior of liquid propellant in the tanks of launch vehicles.

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[in Japanese]

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