## MEMBERS

### Program Members

Numerical simulation is a key tool for analyzing and predicting various phenomena in nature. Nowadays it is also an indispensable tool in industries. For example, the manufacturing industry uses numerical simulations in the design of automobiles, vessels and aircrafts, the construction industry uses it to build high-rise buildings, while the medical industry and nanotechnology industry use it for small objects such as blood flow. Most mathematical models describing these phenomena are expressed by partial differential equations because nature has four independent variables, three in space and one in time. While it is almost impossible to get an analytical solution to these partial differential equations which describe real-world problems, advances in numerical analysis theory and computer performance have made it possible to obtain numerical solutions to these problems, at least to standard ones, at a practical level. Numerical simulation now occupies a major position in computation for science and technology.

Although computers can execute a large amount of the four arithmetic operations at an extremely high speed, the number of operations is finite but not infinite. Because computers cannot process continuously changing independent variables in partial differential equations, they are converted to discrete values, which lead to discrete equations. These discrete equations are then solved using a finite number of operations of addition, subtraction, multiplication, and division. Unlike an analytical solution to partial differential equations, a computer’s solution to discrete equations is no more than an approximate solution. A key issue is to show that when the degree of discretization increases, that is, the increments of space and time become smaller, the approximate solution remains in a bound and converges to the exact solution of the partial differential equations. The above is called stability and convergence, which forms the core of numerical analysis of partial differential equations and guarantees the validity of numerical results. It is important to build a numerical computation scheme to support this discussion. As equations become complex, it is not easy to discuss a whole scheme. However, a theory for constructing schemes suitable to each type of equations has been developed. Mathematical knowledge of partial differential equations and function space is extremely helpful for such a study.

Although numerical simulations have been conducted in different fields often separately, all simulations have commonalities from a view point of numerical analysis of partial differential equations. For example, air flow around an aircraft and blood flow in a body are described using the same type of partial differential equations, and thus, share the same computation scheme and mathematical properties. While it is important to consider characteristics specific to individual phenomenon, mathematics is good at extracting a common set of characteristics from what appear to be different things to provide an understanding of the essence of the problem.

Because improved computer performance has enabled large-scale computations, numerical results are often determined by the quality of a computation scheme. Therefore, it is becoming more important to develop good computation schemes. Toward a complex phenomenon, only a correct understanding of the proper structure and a mathematically sound computation scheme can yield a reliable numerical result.

The objective of numerical analysis of partial differential equations is to obtain a high speed, reliable numerical result using a computer. In practice, my group develops numerical computation schemes, effective algorithms for the scheme, writes codes for the algorithm, and executes the actual computation. In many cases, large-scale simultaneous linear equations must be solved, which makes it necessary to choose an effective solver or to develop and implement a new solver. Hence, numerous procedures are involved in order to obtain numerical simulation results.

Let us show an example of numerical simulation. The figure below depicts a numerical result of multiple bubbles rising in a swayed channel. The motion of each fluid is governed by the Navier-Stokes equations, and surface tension acts at the interface between two fluids. We developed a computation scheme based on an energy-stable finite element method. The figure shows nine bubbles, which were initially at stationary positions, and then moved upwards. The curves in the figure show the stream lines, and the velocity is large at places where the stream lines are thick. Thus, the way all bubbles ultimately merge into one big bubble can be computed.

Figure : Multiple bubbles that rise and merge in a swayed channel.