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Computational design optimization of low-energy buildings

Vala, Jiří (2017)

Proceedings of Equadiff 14

European directives and related national technical standards force the substantial reduction of energy consumption of all types of buildings. This can be done thanks to the massive insulation and the improvement of quality of building enclosures, using the simple evaluation assuming the one-dimensional stationary heat conduction. However, recent applications of advanced materials, structures and technologies force the proper physical, mathematical and computational analysis coming from the thermodynamic...

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

Abdur Rashid, Shakaib Akram (2010)

Applications of Mathematics

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.

Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation

Ujjwal Koley (2012)

Open Mathematics

We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the...

Existence of solutions to nonlinear advection-diffusion equation applied to Burgers' equation using Sinc methods

Kamel Al-Khaled (2014)

Applications of Mathematics

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution...

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

P. E. Vincent, A. Jameson (2011)

Mathematical Modelling of Natural Phenomena

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order...

Fractional-order Bessel functions with various applications

Haniye Dehestani, Yadollah Ordokhani, Mohsen Razzaghi (2019)

Applications of Mathematics

We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between...

Goal oriented a posteriori error estimates for the discontinuous Galerkin method

Dolejší, Vít, Roskovec, Filip (2017)

Programs and Algorithms of Numerical Mathematics

This paper is concerned with goal-oriented a posteriori error estimates for discontinous Galerkin discretizations of linear elliptic boundary value problems. Our approach combines the Dual Weighted Residual method (DWR) with local weighted least-squares reconstruction of the discrete solution. This technique is used not only for controlling the discretization error, but also to track the influence of the algebraic errors. We illustrate the performance of the proposed method by numerical experiments....

Hermite pseudospectral method for nonlinear partial differential equations

Ben-yu Guo, Cheng-long Xu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Hermite polynomial interpolation is investigated. Some approximation results are obtained. As an example, the Burgers equation on the whole line is considered. The stability and the convergence of proposed Hermite pseudospectral scheme are proved strictly. Numerical results are presented.

High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data

Christoph Schwab, Svetlana Tokareva (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data....

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