Takashi Yabe

A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver

Abstract A new numerical method is proposed for general hyperbolic equations. The scheme uses a spatial profile interpolated with a cubic polynomial within a grid cell, and is described in an explicit finite-difference form by assuming that both a physical quantity and its spatial derivative obey the master equation. The method gives stable and less diffusive results even without any flux limiter. It is successfully applied to the KdV equation, a one-dimensional shock-tube problem and a cylindrically converging shock wave.

A universal solver for hyperbolic equations by cubic-polynomial interpolation. II, Two- and three-dimensional solvers

Abstract A new numerical method is proposed for multidimensional hyperbolic equations. The scheme uses a cubic spatial profile within grids, and is described in an explicit finite-difference form by assuming that both the physical quantity and its spatial derivative obey the master equation. The method gives a stable and less diffusive result than the old methods without any flux limiter. Extension to nonlinear equations with nonadvection terms is straightforward.

Unified numerical procedure for compressible and incompressible fluid

A unified numerical procedure is proposed as a solver both for compressible and incompressible fluids. The method approaches the MAC algorithm at very high sound speed and continuously approaches the algorithm for compressible fluid with decreasing sound speed. The advection term is treated by the CIP algorithm which was previously proposed and gives quite accurate and less-diffusive results. This unified procedure is tested both by one-dimensional shock-tube problem and two-dimensional cavity flow at high-Reynolds number.

Description of Complex and Sharp Interface during Shock Wave Interaction with Liquid Drop

A method of treating a sharp discontinuity with a density function is proposed. The surface of the density function is captured within a grid cell throughout the calculation even when the discontinuity surface is largely distorted. This description is made possible by the CIP (cubic-interpolated propagation) method combined with variable transformation. This scheme can be used for detecting a sharp boundary in compressible fluid as well as incompressible fluid. Lower order schemes with the same procedure cannot reproduce the result. This scheme is applied to the interaction of a shock wave with a liquid drop. The liquid mass is conserved within an error of 0.15%.

A Universal Cubic Interpolation Solver for Compressible and Incompressible Fluids

A universal numerical solver commonly usable for compressible and incompressible fluids is proposed. The method approaches the MAC algorithm at very high sound speed and continuously approaches the algorithm for compressible fluid with decreasing sound speed. The advection term is treated by the CIP algorithm which was previously proposed. A single program is applied to one- and two-dimensional shock-tube problems, and two-dimensional liquid flow inside a cavity at high Reynolds number.

Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions

Abstract We have designed and built a special-purpose computer for the reconstruction of virtual images in three dimensions, HORN-1 (HOlographic ReconstructioN). HORN-1 calculates light intensity at high speed. The speed of HORN-1 achieves 300 Mflops-equivalent, and its electric parts cost is 60 000 Japanese yen (500 US dollar). It takes about 24 minutes to reconstruct a 100 × 100 × 8 virtual image from a 400 × 400 hologram.

Description of complex and sharp interface with fixed grids in incompressible and compressible fluid

Abstract A method to treat a sharp discontinuity by the density function is proposed. The surface of the density function is described by one grid throughout the calculation even when the surface is largely distorted. This description is made possible by the CIP method combined with variable transformation. This scheme can be used for detecting sharp boundary of compressible fluid as well as incompressible fluid. Lower order schemes with the same procedure cannot reproduce the result. This scheme is applied to the injection of heavier fluid into lighter fluid and the roll-up of mushroom structure is successfully treated by the density function. The interaction of a shock wave with a liquid drop is an interesting but quite difficult test problem. Even in this case, the liquid mass is conserved within an error of 0.15%.

Holography Machine HORN-1 for Computer-Aided Retrieval of Virtual Three-Dimensional Image

We propose a method to efficiently retrieve the original 3-D (three-dimensional) objects by computers as a virtual image from a hologram. The present method is based on the correlation between the hologram of a point source and the hologram of objects. We have succeeded in the retrieval of objects with the wavelength resolution in any direction. This resolution is not possible with conventional holography. HORN-1 (HOlographic ReconstructioN) is a machine specially prepared for this purpose and enables us to accomplish this work within a short period of time. The machine has achieved a speed of 500 MFLOPS with one circuit.

The compact CIP (cubic-interpolated pseudo-particle) method as a general hyperbolic solver

Abstract A new universal solver is proposed for general hyperbolic equations; multi-dimensional, linear and nonlinear equations with dissipative and dispersive terms. The scheme uses piecewise cubic polynomial interpolation inside meshes. The physical quantity and its spatial derivative are advanced in time according to the given equation. The scheme not only describes a sharp discontinuity with only one mesh but also reproduces the traveling wave train in the dispersive media. The extension to higher dimensions is straightforward.

A method to trace sharp interface of two fluids in calculations involving shocks

A numerical method that can treat a contact discontinuity as a sharp discontinuity is proposed by introducing a density function. The surface of the density function representing the discontinuity can be described with alomost one grid on an Eulerian formulation throughout the calculation even if the surface distortion is large. The algorithm is based on the CIP (cubic-interpolated propagation) method combined with variable transformation. Smearings of the discontinuity are overcome without using any special treatment to the surface. Dependent variables on both sides of the surface are computed in a normal procedure. The present scheme is applied both to a compressible flow and quasi-incompressible flow accompanying shock discontinuities. The result exhibits advantages of this method in robustness and efficiency.