# Numerical Analysis

## New submissions

[ total of 12 entries: 1-12 ]
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### New submissions for Fri, 18 May 18

[1]
Title: Multi-Rate Time Integration on Overset Meshes
Subjects: Numerical Analysis (math.NA)

Overset meshes are an effective tool for the computational fluid dynamic simulation of problems with complex geometries or multiscale spatio-temporal features. When the maximum allowable timestep on one or more meshes is significantly smaller than on the remaining meshes, standard explicit time integrators impose inefficiencies for time-accurate calculations by requiring that all meshes advance with the smallest timestep. With the targeted use of multi-rate time integrators, separate meshes can be time-marched at independent rates to avoid wasteful computation while maintaining accuracy and stability. This work applies time-explicit multi-rate integrators to the simulation of the compressible Navier-Stokes equations discretized on overset meshes using summation-by-parts (SBP) operators and simultaneous approximation term (SAT) boundary conditions. We introduce a novel class of multi-rate Adams-Bashforth (MRAB) schemes that offer significant stability improvements and computational efficiencies for SBP-SAT methods. We present numerical results that confirm the numerical efficacy of MRAB integrators, outline a number of outstanding implementation challenges, and demonstrate a reduction in computational cost enabled by MRAB. We also investigate the use of our method in the setting of a large-scale distributed-memory parallel implementation where we discuss concerns involving load balancing and communication efficiency.

[2]
Title: Generalized least square homotopy perturbations for system of fractional partial differential equations
Subjects: Numerical Analysis (math.NA)

In this paper, generalized aspects of least square homotopy perturbations are explored to treat the system of non-linear fractional partial differential equations and the method is called as generalized least square homotopy perturbations (GLSHP). The concept of partial fractional Wronskian is introduced to detect the linear independence of functions depending on more than one variable through Caputo fractional calculus. General theorem related to Wronskian is also proved. It is found that solutions converge more rapidly through GLSHP in comparison to classical fractional homotopy perturbations. Results of this generalization are validated by taking examples from nonlinear fractional wave equations.

[3]
Title: Method of improvement of convergence Fourier series and interpoliation polynomials in orthogonal functions
Subjects: Numerical Analysis (math.NA)

There is proposed a method for improving the convergence of Fourier series by function systems, orthogonal at the segment, the application of which allows for smooth functions to receive uniformly convergent series. There is also proposed the method of phantom nodes improving the convergence of interpolation polynomials on systems of orthogonal functions, the application of which in many cases can significantly reduce the interpolation errors of these polynomials. The results of calculations are given at test cases using the proposed methods for trigonometric Fourier series; these calculations illustrate the high efficiency of these methods. Undoubtedly, the proposed method of phantom knots requires further theoretical studies.

[4]
Title: A convergent evolving finite element algorithm for mean curvature flow of closed surfaces
Subjects: Numerical Analysis (math.NA)

A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in {Dziuk's} method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix--vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

[5]
Title: Numerical conservation of energy, momentum and actions for extended RKN methods when applied to nonlinear wave equations via spatial spectral semi-discretizations
Authors: Bin Wang, Xinyuan Wu
Subjects: Numerical Analysis (math.NA)

This paper analyses the long-time behaviour of extended Runge--Kutta--Nystr\"{o}m (ERKN) methods when applied to nonlinear wave equations. It is shown that energy, momentum, and all harmonic actions are approximately preserved over a long time for a one-stage explicit ERKN method when applied to nonlinear wave equations via spectral semi-discretisations. The results are proved by deriving a multi-frequency modulated Fourier expansion of the ERKN method and showing three almost-invariants of the modulation system.

[6]
Title: Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations
Comments: 20 pages, 4 figures, 8 tables
Subjects: Numerical Analysis (math.NA)

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.

[7]
Title: Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II
Subjects: Numerical Analysis (math.NA)

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size $h$ and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness using a non conforming right hand side. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed $H(\operatorname{div})$-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements [P. L. Lederer, J. Sch\"oberl, IMA Journal of Numerical Analysis, 2017] and is derived by a direct approach instead of using a best approximation C\'{e}a like result. We further treat the impact of the reconstruction operator on the $hp$ analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier--Stokes equations which is based on the methods recently presented in [C. Lehrenfeld, J. Sch\"oberl, \emph{Comp. Meth. Appl. Mech. Eng.}, 361 (2016)] and includes the ideas of the reconstruction operator.

[8]
Title: Open canals flow with fluvial to torrential phase transitions on networks
Subjects: Numerical Analysis (math.NA)

Network flows and specifically open canal flows can be modeled by systems of balance laws defined on topological graphs. The shallow water or Saint-Venant system of balance laws is one of the most used model and present two phases: fluvial or sub-critical and torrential or super critical. Phase transitions may occur within the same canal but transitions related to networks are less investigated. In this paper we provide a complete characterization of possible phase transitions for a simple network with two canals and one junction. Our analysis allows the study of more complicate scenarios. Moreover, we provide some numerical simulations to show the theory at work.

[9]
Title: Saddle Point Least Squares Preconditioning of Mixed Methods
Comments: Submitted to CAMWA on 5/17/18
Subjects: Numerical Analysis (math.NA)

We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided. For the proposed discrete spaces and solvers, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete $\inf-\sup$ and $\sup-\sup$ constants of the pair of discrete spaces.

### Cross-lists for Fri, 18 May 18

[10]  arXiv:1805.06842 (cross-list from physics.flu-dyn) [pdf, ps, other]
Title: Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations
Authors: Yue Pu (CNA), Robert Pego (CNA), Denys Dutykh (LAMA), Didier Clamond (JAD)
Comments: 25 pages, 4 figures, 23 references. Accepted to Comm. Math. Sci. Other author's papers can be downloaded at this http URL
Subjects: Fluid Dynamics (physics.flu-dyn); Analysis of PDEs (math.AP); Numerical Analysis (math.NA)

We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

### Replacements for Fri, 18 May 18

[11]  arXiv:1710.02547 (replaced) [pdf, other]
Title: An isogeometric finite element formulation for phase transitions on deforming surfaces
Comments: Restructured Sec. 2 - Sec. 5; refined and updated the computational results; added appendix A, B and C
Subjects: Numerical Analysis (math.NA)
[12]  arXiv:1610.03883 (replaced) [pdf, ps, other]
Title: A method for obtaining Fibonacci identities
Journal-ref: Integers, Vol. 18 (2018), #A42
Subjects: Number Theory (math.NT); Information Theory (cs.IT); Numerical Analysis (math.NA); Rings and Algebras (math.RA)
[ total of 12 entries: 1-12 ]
[ showing up to 2000 entries per page: fewer | more ]

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