اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNumerical study on the effect of geometric approximation error in the numerical solution of PDEs using a high-order curvilinear mesh
101
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. This paper aims to investigate the effect of geometric approximation error by using a high-order mesh with negligible geometric approximation error, even for high order polynomial of order p. To achieve this goal, the high-order mesh generator from CAD geometry called NekMesh is adapted for surface mesh generation in comparison to traditional meshes with non-negligible geometric approximation error. Two types of numerical tests are considered. Firstly, the accuracy of differential operators is compared for various p on a curved element of the sphere. Secondly, by applying the method of moving frames, four different time-dependent PDEs on the sphere are numerically solved to investigate the impact of geometric approximation error on the accuracy and conservation properties of high-order numerical schemes for PDEs on the sphere.
قيم البحث
اقرأ أيضاً
We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value pr
oblem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.
In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partia
l differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.
In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) a
pproximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.
We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions $d=2$ and $3$. This requires to approximate first the scattering field, for some incident waves, which
can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by G. Vainikko in cite{V} to solve such equations when considering the Lame operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.
In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK m
odel and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formulation of the $L^1$-Optimal Transport Problem. We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in $REAL^3$ using the Surface Finite Element Model approach to approximate the Laplace-Beltrami equation arising from the model. We test the accuracy and efficiency of the proposed numerical scheme, comparing our approximate solution with respect to an exact solution on a 2D sphere. The results show that the numerical scheme is efficient, robust, and more accurate with respect to other numerical schemes presented in the literature for the solution of ls$L^1$-Optimal Transport Problem on 2D surfaces.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
