6. 2 Linearized Expressions

The main aim of the linearized expression approach is to ease the specification of discretized differential expressions for numerical computation. During the implementation of a specific physical model, the most error prone as well as time consuming part is the linearization of the expression which is required to obtain the system matrix. For this reason, many different derivatives of the discretized expression have to be calculated and implemented.

One of the major advantages of the linearized expressions is that they can be directly used for discretization schemes. While the abstract formulation of the equation presented in Chapter 3 shows the principles of calculating derivatives of functions which are based on topological complexes, the use of linearized expressions enables the calculation of the respective system matrices for the given problems.

One overcomes these problems by implementing only one linear functional data structure which comprises the linear functional dependence of the discretized expressions on the solution variables in the neighborhood of a linearization point or vector. Such an algebraic structure can be easily implemented by explicitly defining basic arithmetic operations such as addition, multiplication and function application.

As a result, equation systems and system matrices can be derived automatically only by the specification and evaluation of a discretized residual expression. Derivatives with respect to the independent variables are implicitly available and do not have to be calculated explicitly by hand. Therefore, the implementation effort can be reduced enormously and testing and validation of a model can be simplified drastically.

The approach for linearized expressions can of course be used for arbitrary expansions with respect to many different variables. As an example, one can use this approach to specify higher-order Taylor series and explicitly store second order derivatives, or use other than polynomial functions. As long as the formulation describes a linear function space, and special operations preserve the structure (e.g. differentiation for Fourier series), the obtained data structure can be used for many different purposes.

By the possibility to formulate linearized expressions in a functional manner one can specify discretized differential equations in a straightforward manner, while the effort for the specification is reduced to a minimum. Consequently, expressions can be written concisely and are expanded and differentiated automatically via the specification environment. BibliographyBibliography

Michael 2008-01-16