Solution of Evolutionary Partial Differential Equations Using Adaptive Finite Differences with Pseudospectral Post-processing

1997 ◽  
Vol 131 (2) ◽  
pp. 280-298 ◽  
Author(s):  
L.S. Mulholland ◽  
Y. Qiu ◽  
D.M. Sloan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Differences ◽  
Post Processing ◽  
Partial Differential

Client-server architecture for pre and post-processing of real problems involving two-dimensional generalized coordinates

2015 ◽  
Vol 11 (2) ◽  
pp. 226-245
Author(s):  
José Luiz Vilas Boas ◽  
Fabio Takeshi Matsunaga ◽  
Neyva Maria Lopes Romeiro ◽  
Jacques Duílio Brancher
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Post Processing ◽  
Coordinate Systems ◽  
Partial Differential ◽  
Content Type ◽  
Client Server ◽  
Web System ◽  
The Web

Purpose – The aim of this paper is to propose a Web environment for pre-processing and post-processing for 2D problems in generalized coordinate systems. Design/methodology/approach – The system consists of a Web service for client-server communication, a database for user information, simulation requests and results storage, a module of (for) calculation processing (front-end) and a graphical interface for visualization of discretized mesh (back-end). Findings – The Web system was able to model real problems and situations, where the user can describe the problem or upload a geometry file descriptor, generated from computer graphics software. The Web system, programmed for finite difference solutions, was able to generate a mesh from other complex methods, such as finite elements method, adapting it to the proposed Web system, respecting the finite difference mesh structure. Research limitations/implications – The proposed Web system is limited to solve partial differential equations by finite difference discretization. We need to study about refinement and parameters adaptations to solve partial differential equations simulated with other methods. Practical implications – The Web system includes implications for the development of a powerful real problems simulator, which is useful for computational physics researchers and engineers. The Web system uses several technologies, such as Primefaces, JavaScript, JQuery and HTML, to provide an interactive user interface. Originality/value – The main contribution of this work is the availability of a generic Web architecture for including other types of coordinate systems and to solve others partial differential equations. Moreover, this paper presents an extended version of the work presented in ICCSA 2014.

Poroelastic aspects in geothermics

2020 ◽  
Author(s):  
Bianca Kretz ◽  
Willi Freeden ◽  
Volker Michel
Keyword(s):  
Partial Differential Equations ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Fundamental Solutions ◽  
Multiscale Methods ◽  
Method Of Fundamental Solutions ◽  
Post Processing ◽  
Partial Differential ◽  
D’Alembert Equation ◽  
Phd Thesis

<p>The aspect of poroelasticity is anywhere interesting where a solid material and a fluid come into play and have an effect on each other. This is the case in many applications and we want to focus on geothermics. It is useful to consider this aspect since the replacement of the water in the reservoir below the Earth's surface has an effect on the sorrounding material and vice versa. The underlying physical processes can be described by partial differential equations, called the quasistatic equations of poroelasticity (QEP). From a mathematical point of view, we have a set of three (for two space and one time dimension) partial differential equations with the unknowns u (displacement) and p (pore pressure) depending on the space and the time.</p><p>Our aim is to do a decomposition of the data given for u and p in order that we can see underlying structures in the different decomposition scales that cannot be seen in the whole data.<br>For this process, we need the fundamental solution tensor of the QEP (cf. [1],[5]).<br>That means we assume that we have given data for u and p (they can be obtained for example by a method of fundamental solutions, cf. [1]) and want to investigate a post-processing method to these data. Here we follow the basic approaches for the Laplace-, Helmholtz- and d'Alembert equation (cf. [2],[4],[6]) and the  Cauchy-Navier equation as a tensor-valued ansatz (cf. [3]). That means we want to modify our elements of the fundamental solution tensor in such a way that we smooth the singularity concerning a parameter set τ=(τ<sub>x</sub>,τ<sub>t</sub>). <br>With the help of these modified functions, we construct scaling functions which have to fulfil the properties of an approximate identity.<br>They are convolved with the given data to extract more details of u and p.</p><p><strong>References</strong></p><p>[1] M. Augustin: A method of fundamental solutions in poroelasticity to model the stress field in geothermal reservoirs, PhD Thesis, University of Kaiserslautern, 2015, Birkhäuser, New York, 2015.<br>[2] C. Blick, Multiscale potential methods in geothermal research: decorrelation reflected post-processing and locally based inversion, PhD Thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015.<br>[3] C. Blick, S. Eberle, Multiscale density decorrelation by Cauchy-Navier wavelets, Int. J. Geomath. 10, 2019, article 24.<br>[4] C. Blick, W. Freeden, H. Nutz: Feature extraction of geological signatures by multiscale gravimetry. Int. J. Geomath. 8: 57-83, 2017.<br>[5] A.H.D. Cheng and E. Detournay: On singular integral equations and fundamental solutions of poroelasticity. Int. J. Solid. Struct. 35, 4521-4555, 1998.<br>[6] W. Freeden, C. Blick: Signal decorrelation by means of multiscale methods, World of Mining, 65(5):304-317, 2013.<br><br></p>

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