Transformation approach to numerically integrating PDEs by means of WDF principles

Abstract

Abstract. As has been shown, attractive methods for numerically integrating partial differential equations (PDEs) resulting from physical problems can be obtained by simulating the actual physical passive (conservation of energy) dynamical system by means of a discrete passive dynamical system, and this in such a way that the full parallelism and the exclusively local nature of the interconnections (principle of action at proximity) are preserved. An alternative approach for developing such methods is presented which, while still using principles of the same type as those on which multidimensional wave digital filters (WDFs) are based, involves appropriate transformations of the original coordinates of the physical problem at hand. This alternative approach is not only easier to apply than the one referred to above but also more general; it is illustrated on the one hand by the same examples as those that have been used for the other approach, and on the other by showing the applicability to Maxwell's equations.

Analysis of Parallel Multidimensional Wave Digital Filtering Network on IBM Cell Broadband Engine

“…Applying the standard procedure known from the MD wave digital filtering [4,5] for transforming the set of PDEs to its equivalent discrete passive model, a lumped MD-passive Kirchhoff circuit (MDKC) and its discrete approximation of MDWDF network [11] depicted in Figures 1(a) and 1(b), respectively, are obtained for the numerical integration of the LSW system. Since the resulting network behaves in the same way as the continuous one, it also preserves passivity for the discrete dynamical system, thus ensuring full robustness and stability of the algorithm [4,12].…”

“…Since the resulting network behaves in the same way as the continuous one, it also preserves passivity for the discrete dynamical system, thus ensuring full robustness and stability of the algorithm [4,12]. The reader is referred to [8] for more details of converting the given physical system to form the MDWDF network via the MDKC.…”

“…Built on properties of the traveling wave formulation of lumped electrical elements to the modeling and simulation of a system represented by PDEs, a novel approach named wave digital filtering (WDF) network [3] had been proposed in the past due to its excellent features that fit requirements of practical interest [4,5]. Unlike most types of digital filter, every delay element in a WDF network can be interpreted physically as holding the current state of a mass or spring (or capacitor or inductors) [3].…”

Accommodating Linear and Nonlinear Boundary Conditions in Wave Digital Simulations of PDE Systems

“…where A and P are nn matrix, d, f, s and q are column n-vectors, and e is a scalar. A, P, d, f, s and e may befunctions of the spatial variables t 1 and t 2 , but in what follows, we will take these coe cients to betime-invariant. Only the vector of sources, q, will beallowed to be time-varying.…”

Discretization schemes and numerical approximations of PDE impainting models and a comparative evaluation on novel real world MRI reconstruction applications

“…An alternative approach for simulating and solving PDEs in discrete space-time has been proposed by Fettweis and is based on properties of Kirchhoff networks [33]- [37]. This technique involves firstly finding a multi-dimensional lumped electrical network which represents the behavior of the linear or non-linear system.…”