Numerical methods for partial differential equations: algorithms and software on innovative computer architectures

我们的世界是由物理和技术现象、物体和情况组成的。为了研究世界，经常使用数学模型。这些通常是方程，例如，常微分方程式或偏微分方程式、线性或非线性方程式、积分方程式等。这些数学模型往往无法用解析数学技巧来求解。用数值方法求解或近似求解数学模型。这些方法是在计算机上实现和运行的。数值方法和计算都可能引入误差。对计算机结果进行了研究，并对可能的误差来源进行了调查。如果结果不令人满意，则必须调整数学模型或求解模型的数值方法，或者调整实现该方法的计算机程序，并重复整个过程。数学建模、发展求解数学模型的数值方法以及在计算机上实施这些方法是跨越计算机科学和数学以及物理学和工程学的重要和广泛的研究领域。

Our world is comprised of physical and technological phenomena, objects and situations. In order to study the world, mathematical models are often used. These are usually equations, for example, ordinary or partial differential equations, linear or non-linear equations, integral equations,etc. The mathematical models are often insolvable by analytic mathematical techniques. Numerical methods are applied to solve or approximate the solution to mathematical models. These methods are implemented and run on computers. Numerical methods as well as computations may introduce errors. The computer results are studied and possible sources of errors are investigated. If the results are not satisfactory, either the mathematical model, or the numerical method to solve the model, or the computer program implementing the method have to be adjusted and the whole procedure repeated. Mathematical modelling, the development of numerical methods to solve mathematical models and the implementation of the methods on computers are important and wide areas of research, spanning computer science and mathematics, as well as physics and engineering.