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

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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.