China
Africa
Explore chapters and articles related to this topic
Infinite Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Lemma 7.5.8 (Summation by Parts). Let {an} and {bn} be two sequences of real numbers, let m ∈ N, and let Bn= Σk=1nbk for all n ∈ N. then∑k=nmakbk=am+1Bm−anBn−1−∑k=nm(ak+1−ak)Bk.
Finite-Difference Schemes for Partial Differential Equations
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
Next, we notice that am+1um+1 = amum+1 + (am+1 − am)um+1. Substituting this expression into the previous formula, we again perform summation by parts, which is analogous to the continuous integration by parts, and which yields:
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
To prove convergence of the series, we begin with summation by parts, an algebraic formula analogous to integration by parts. The partial sums BN in Lemma 3.2 are analogous to integrals, and the differences an + 1 − an are analogous to derivatives.
Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media
Published in Applicable Analysis, 2022
Nitesh Verma, Bryan Gómez-Vargas, Luis Miguel De Oliveira Vilaca, Sarvesh Kumar, Ricardo Ruiz-Baier
We proceed similarly to the proof of Lemmas 2.1 and 2.4. We focus first on the stability of (62)–(64). Taking in (62), using Cauchy-Schwarz inequality, applying Young's inequality with constants chosen conveniently, and then, summing over and multiplying by , we readily get (68), where is a constant depending on , and τ. Now, in Equations (63) and (64), we take and , respectively, to obtain Thus, applying Young's inequality to the first and second term, and summation by parts to the last term, on the right-hand side of (70), we obtain (67).
Adjoint-based shape optimization for the minimization of flow-induced hemolysis in biomedical applications
Published in Engineering Applications of Computational Fluid Mechanics, 2021
Georgios Bletsos, Niklas Kühl, Thomas Rung
The numerical procedure for the solution of the primal and adjoint system is based upon the Finite Volume Method (FVM) employed by FreSCo (Rung et al., 2009). Analogue to the use of integration-by-parts in deriving the continuous adjoint equations (Kühl et al., 2019, 2021), summation-by-parts is employed to derive the building blocks of the discrete adjoint expressions. A detailed derivation of this hybrid adjoint approach can be found in Kröger et al. (2018), Kühl, Kröger et al. (2021) and Stück and Rung (2013). The last two terms of the adjoint momentum equation, involving hemolysis contributions, are added explicitly to the RHS. The segregated algorithm uses a cell-centered, collocated storage arrangement for all transport properties. The implicit numerical approximation is second order accurate in space and supports polyhedral cells. Both, the primal and adjoint pressure–velocity coupling is based on a SIMPLE method and possible parallelization is realized by means of a domain decomposition approach (Yakubov et al., 2013, 2015).
Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods
Published in International Journal of Computational Fluid Dynamics, 2019
Philipp Birken, Gregor J. Gassner, Lea M. Versbach
The goal of our research is the construction of efficient Jacobian-free preconditioners for high order Discontinuous Galerkin (DG) discretisations with implicit time integration. One of our main interests is three-dimensional unsteady compressible flow. High-order DG methods (and related methods such as Flux Reconstruction (FR) discretisations) offer great potential for Large Eddy Simulation (LES) of turbulent flows with geometries, such as jet engines. The idea of DG (or FR) is to approximate the solution element-wise using a polynomial, which is allowed to be discontinuous across element interfaces, see Kopriva (2009) and Huynh (2007). Communication and coupling of degrees of freedom (DOF) is only across faces, whereas the element-local computations are very dense. As a result, DG methods are very well suited for domain-decomposition-based parallelisation (see, e.g. Hindenlang et al. 2012; Vincent et al. 2016). The specific variant we consider is the DG Spectral Element Method (DG-SEM), e.g. Kopriva, Woodruff, and Hussaini (2002). We use a Lagrange-type (nodal) basis with Gauss–Lobatto (GL) quadrature nodes with the collocation of the discrete integration. These choices yield DG operators that satisfy the summation-by-parts (SBP) property (see Gassner 2013), which is the discrete analogue to integration-by-parts. SBP is key to construct methods that are discretely entropy stable and/or kinetic energy preserving.