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Formal Proof of a Wave Equation Resolution Scheme: The Method Error [chapter]

Sylvie Boldo, François Clément, Jean-Christophe Filliâtre, Micaela Mayero, Guillaume Melquiond, Pierre Weis
2010 Lecture Notes in Computer Science  
Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent.  ...  We present a comprehensive formalization of the simplest one and formally prove its convergence in Coq.  ...  INRIA inria-00450789, version 2 -5 May 2010 Formal Proof of a Wave Equation Resolution Scheme: the Method Error Variables x and ∆x are restricted to subsets S and P of R 2 .  ... 
doi:10.1007/978-3-642-14052-5_12 fatcat:yxfxxwktbrezvcbck62gouiuhu

Formal Proof of a Wave Equation Resolution Scheme: the Method Error [article]

Sylvie Boldo, François Clément, Jean-Christophe Filliâtre (INRIA Saclay - Ile de France, LRI), Micaela Mayero (LIPN, INRIA Rhône-Alpes / LIP Laboratoire de l'Informatique du Parallélisme), Guillaume Melquiond (INRIA Saclay - Ile de France, LRI), Pierre Weis
2011 arXiv   pre-print
Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent.  ...  We present a comprehensive formalization of the simplest one and formally prove its convergence in Coq.  ...  As a consequence, the proof of the method error has to be combined INRIA with a proof on the rounding error, in order to get a full-fledged correction proof.  ... 
arXiv:1001.4898v3 fatcat:floje32nkvh3naqr6e4nhczt6i

A limiting strategy for the back and forth error compensation and correction method for solving advection equations

Lili Hu, Yao Li, Yingjie Liu
2016 Mathematics of Computation  
We further study the properties of the back and forth error compensation and correction (BFECC) method for advection equations such as those related to the level set method and for solving Hamilton-Jacobi  ...  Typically, a formal second order method method for solving a time dependent Hamilton-Jacobi equation requires quadratic interpolation in space.  ...  error of the underlying scheme.  ... 
doi:10.1090/mcom/3026 fatcat:uoaqcmcfurc4nmd6kqjpzx5leq

Error Analysis of Splitting Methods for the Time Dependent Schrödinger Equation

Sergio Blanes, Fernando Casas, Ander Murua
2011 SIAM Journal on Scientific Computing  
We carry out an error analysis of these integrators and propose a strategy which allows us to construct different splitting symplectic methods of different order (even of order zero) possessing a large  ...  A typical procedure to integrate numerically the time dependent Schrödinger equation involves two stages. In the first one carries out a space discretization of the continuous problem.  ...  the formalism can also be adapted to an odd number of points).  ... 
doi:10.1137/100794535 fatcat:h26dpcnd5vcqdmehcv5zlvt6cy

Operator Splitting Methods with Error Estimator and Adaptive Time-Stepping. Application to the Simulation of Combustion Phenomena [chapter]

Stéphane Descombes, Max Duarte, Marc Massot
2016 Splitting Methods in Communication, Imaging, Science, and Engineering  
Nevertheless, splitting errors are introduced in the numerical approximations due to the separate evolution of the split subproblems and can compromise a reliable reproduction of the coupled dynamics.  ...  A multi-physics problem is thus split in subproblems of different nature with a significant reduction of the algorithmic complexity and computational requirements of the numerical solvers.  ...  The splitting error in the numerical solution is therefore overestimated since the S-scheme is formally of higher order.  ... 
doi:10.1007/978-3-319-41589-5_19 fatcat:uenwq36qsbagnplnuyjv6fyr2m

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

WeiZhu Bao, YongYong Cai, XiaoWei Jia, Jia Yin
2016 Science China Mathematics  
We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter  ...  Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential  ...  Part of this work was done when the authors were visiting the Institute for Mathematical Sciences at the National University of Singapore in 2015.  ... 
doi:10.1007/s11425-016-0272-y fatcat:mxvmzk2z3jf7fgzg4gtz46jsbi

An Error Analysis of Conservative Space-Time Mesh Refinement Methods for the One-Dimensional Wave Equation

Patrick Joly, Jerónimo Rodriguez
2005 SIAM Journal on Numerical Analysis  
In this paper, we show the L 2 convergence of these schemes and provide optimal error estimates. The proof is based on energy techniques and bootstrap arguments.  ...  The stability of such methods is guaranteed by construction through the conservation of a discrete energy.  ...  However, it is not difficult to see that the proofs of section 4 (based on energy methods) can be adapted to the case of the 1D wave equation with spatially variable coefficients. Remark 3.3.  ... 
doi:10.1137/040603437 fatcat:32lg7ajlqzhhvcpjac7pz2m5ye

The error-minimization-based rezone strategy for arbitrary Lagrangian-Eulerian methods

Konstantin Lipnikov, Mikhail Shashkov
2006 Numerical Methods for Partial Differential Equations  
In this article, we describe a new rezone strategy which minimizes the L 2 norm of the solution error and maintains smoothness of the mesh.  ...  The efficiency of the new method is demonstrated with numerical experiments.  ...  This error is a superposition of the interpolation error, an error due to a time integration method and the space discretization error.  ... 
doi:10.1002/num.20113 fatcat:vn7kxs2fsjds3csa6ock7rqyia

The Fast Multipole Method I: Error Analysis and Asymptotic Complexity

Eric Darve
2000 SIAM Journal on Numerical Analysis  
This paper is concerned with the application of the fast multipole method (FMM) to the Maxwell equations.  ...  This leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles. Emphasis here is on a rigorous mathematical approach to the problem.  ...  Using an iterative method (conjugate gradient, GMRES . . . ) and a preconditioner (such as SPAI; see [16] and [2] ) the computer intensive part of the resolution of the linear problem reduces to the  ... 
doi:10.1137/s0036142999330379 fatcat:citpks3ghbc2nlxssqbnpc2xby

Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation [article]

Weizhu Bao, Yongyong Cai, Yue Feng
2022 arXiv   pre-print
We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with  ...  and energy method, an improved uniform error bound at O(h^m-1 + ετ^2) is established in H^1-norm for the long-time dynamics up to the time at O(1/ε) of the Schrödinger equation with O(ε)-potential with  ...  Acknowledgement The authors would like to thank the anonymous referee for the invaluable comments and suggestions.  ... 
arXiv:2109.08940v2 fatcat:qhxibdpovjawbppexm3nvui2iq

Error-Correcting Neural Networks for Semi-Lagrangian Advection in the Level-Set Method [article]

Luis Ángel Larios-Cárdenas, Frédéric Gibou
2021 arXiv   pre-print
In passive advection problems, for example, our method can achieve the same precision as the baseline scheme at twice the resolution but at a fraction of the cost.  ...  We present a machine learning framework that blends image super-resolution technologies with scalar transport in the level-set method.  ...  Acknowledgements Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC).  ... 
arXiv:2110.11611v1 fatcat:ms5iqfjrnzah5imvhxd5fxy2zy

A Priori Error Analysis of the Finite Element Heterogeneous Multiscale Method for the Wave Equation over Long Time

Assyr Abdulle, Timothée Pouchon
2016 SIAM Journal on Numerical Analysis  
Key words. a priori error analysis, multiscale method, heterogeneous media, effective equations, wave equation, long time behavior, dispersive waves AMS subject classifications. 65M60, 65N30, (74Q10, 74Q15  ...  on the ε-scale (see [13] for the specific case of the wave equation).  ...  This method was shown to be wellposed and consistent with the classical homogenization problem for the wave equation.  ... 
doi:10.1137/15m1025633 fatcat:gctwflyetfho3k23feoiym7boi

Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

Erik Burman
2015 Mathematical Models and Methods in Applied Sciences  
In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second order strong stability preserving Runge-Kutta method.  ...  Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory  ...  Proof. For a proof we refer to Refs. 17, 18. First order schemes We will first consider the artificial viscosity method obtained using the definition (3.2) in (3.5). Proposition 5.1.  ... 
doi:10.1142/s0218202515500517 fatcat:dklyi34zprhjpd2sjlyoozxb4m

Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study

T. Duretz, D. A. May, T. V. Gerya, P. J. Tackley
2011 Geochemistry Geophysics Geosystems  
The use of a more local interpolation scheme (1-cell) decreases the absolute velocity and pressure discretization errors.  ...  In this study, we numerically investigate the discretization errors and order of accuracy of the velocity and pressure solution obtained from the FD-MIC scheme using two-dimensional analytic solutions.  ...  Acknowledgments [62] The authors thank Harro Schmeling and Cedric Thieulot for constructive reviews that improved the quality and clarity of the paper.  ... 
doi:10.1029/2011gc003567 fatcat:3zjnhjmtjvgtvl757hu6nbizra

Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics

Othmar Koch, Christof Neuhauser, Mechthild Thalhammer
2013 Mathematical Modelling and Numerical Analysis  
In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated.  ...  The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives.  ...  A local error expansion In this section, we deduce a local error expansion of high-order exponential operator splitting methods applied to a nonlinear evolutionary problem of the form Employing the formal  ... 
doi:10.1051/m2an/2013067 fatcat:mjvkb2sw2zg7rjjvunz5w6ayua
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