I am interested in understanding phenomena in nature and industry, for example, water flow in rivers and heat transfer in furnaces, by numerical simulations. The numerical simulation consists of three parts: mathematical modeling, discretization, and numerical computation. First, in the mathematical modeling part, the phenomena are described by differential equations based on physical laws. Next, in the discretization part, the differential equations are approximated by linear systems, which can be handled with computers. Finally, in the numerical computation part, methods to solve the linear systems are implemented in computers to show the phenomena. I focus my attention to proposed numerical simulation methods and to justify the accuracy and efficiency of the methods mathematically. Moreover, I perform numerical computations of various practical problems by using the proposed methods, then I apply them for understanding the phenomena in nature and the design of industrial products.