A Laplace Transform Certified Reduced Basis Method; Application to the Heat Equation and Wave Equation
نویسندگان
چکیده
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline-Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach. To cite this article: DBP Huynh, DJ Knezevic, AT Patera, C. R. Acad. Sci. Paris, Ser. I XXX (2010). Résumé Une méthode de bases réduites certifiée utilisant la transformée de Laplace ; Application l’équation de la chaleur et l’équation des ondes Nous introduisons une méthode de bases réduites certifiée pour l’équation de la chaleur et l’équation des ondes utilisant la transformée de Laplace. Les ingrédients essentiels sont les suivants : une approximation par bases réduites certifiée de la transformée de Laplace, une transformée de Laplace inverse pour l’approximation de l’output par bases réduites en temps et l’établissement de bornes d’erreur correspondantes rigoureuses, un filtre en temps (de Butterworth) pour mettre en place la troncation “modale” nécessaire, une décomposition en fonctions propres par bases réduites et une intégrale de contour pour la décomposition Offline-Online. Nous présentons des résultats numériques qui démontrent la précision et l’éfficacité de l’approche. Pour citer cet article : DBP Huynh, DJ Knezevic, AT Patera, C. R. Acad. Sci. Paris, Ser. I XXX (2010). Email addresses: [email protected] (DBP Huynh), [email protected] (DJ Knezevic), [email protected] (AT Patera). Preprint submitted to the Académie des sciences 21 février 2011 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 2011 2. REPORT TYPE 3. DATES COVERED 00-00-2011 to 00-00-2011 4. TITLE AND SUBTITLE A Laplace Transform Certified Reduced Basis Method; Application to the Heat Equation and Wave Equation 5a. CONTRACT NUMBER
منابع مشابه
Analytical Analysis of The Dual-phase-lag Heat Transfer Equation in a Finite Slab with Periodic Surface Heat Flux (RESEARCH NOTE)
This work uses the dual-phase-lag (DPL) model of heat conduction to demonstrate the effect of temperature gradient relaxation time on the result of non-Fourier hyperbolic conduction in a finite slab subjected to a periodic thermal disturbance. DPL model combines the wave features of hyperbolic conduction with a diffusion-like feature of the evidence not captured by the hyperbolic case. For the ...
متن کاملNumerical solution of time-dependent foam drainage equation (FDE)
Reduced Differental Transform Method (RDTM), which is one of the useful and effective numerical method, is applied to solve nonlinear time-dependent Foam Drainage Equation (FDE) with different initial conditions. We compare our method with the famous Adomian Decomposition and Laplace Decomposition Methods. The obtained results demonstrated that RDTM is a powerful tool for solving nonlinear part...
متن کاملDifferential Transform Method to two-dimensional non-linear wave equation
In this paper, an analytic solution is presented using differential transform method (DTM) for a class of wave equation. The emphasis is on the nonlinear two-dimensional wave equation. The procedures introduced in this paper are in recursive forms which can be used to obtain the closed form of the solutions, if they are required. The method is tested on various examples, and the results reveal ...
متن کاملModeling Diffusion to Thermal Wave Heat Propagation by Using Fractional Heat Conduction Constitutive Model
Based on the recently introduced fractional Taylor’s formula, a fractional heat conduction constitutive equation is formulated by expanding the single-phase lag model using the fractional Taylor’s formula. Combining with the energy balance equation, the derived fractional heat conduction equation has been shown to be capable of modeling diffusion-to-Thermal wave behavior of heat propagation by ...
متن کاملNon-Fourier heat conduction equation in a sphere; comparison of variational method and inverse Laplace transformation with exact solution
Small scale thermal devices, such as micro heater, have led researchers to consider more accurate models of heat in thermal systems. Moreover, biological applications of heat transfer such as simulation of temperature field in laser surgery is another pathway which urges us to re-examine thermal systems with modern ones. Non-Fourier heat transfer overcomes some shortcomings of Fourier heat tran...
متن کامل