An implicit-explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

We construct and analyze a second-order implicit-explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate

A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions

In this thesis, a unified error analysis for discretizations of nonlinear first- and second-order wave-type equations is provided. For this, the wave equations as well as their space discretizations are considered as nonlinear evolution equations in Hilbert spaces. The space discretizations are supplemented with Runge-Kutta time discretizations. By employing stability properties of monotone operators, abstract error bounds for the space, time, and full discretizations are derived. Further, for semilinear second-order wave-type equations, an implicit-explicit time integration scheme is presented. This scheme only requires the solution of a linear system of equations in each time step and it is stable under a step size restriction only depending on the nonlinearity. It is proven that the scheme converges with second order in time and in combination with the abstract space discretization of the unified error analysis, corresponding full discretization error bounds are derived. The abstract results are used to derive convergence rates for an isoparametric finite element space discretization of a wave equation with kinetic boundary conditions and nonlinear forcing and damping terms. For the combination of the finite element discretization with Runge-Kutta methods or the implicit-explicit scheme, respectively, error bounds of the resulting fully discrete schemes are proven. The theoretical results are illustrated by numerical experiments

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods.Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287–301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.The authors would like to thank Prof. Eugenio Oñate (U. Politècnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.) Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSCCNS) and Universitat Politècnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola Energías Renovables, and by grant N62909-09-1-4083 of the Office of Naval Research Global. At NPS, SM, AM, MK, and FXG were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. SM, MK, and AM are grateful to the National Research Council of the National Academies.Peer ReviewedPostprint (author's final draft

TIME FILTERS FOR NUMERICAL WEATHER PREDICTION

The Robert-Asselin (RA) time filter combined with leapfrog scheme is widely used in numerical models of weather and climate. The RA filter suppresses the spurious computational mode associated with the leapfrog method, and successfully stabilizes the numerical solution. However, it also weakly dampens the physical mode and degrades the formal second-order accuracy of the leapfrog scheme to first order. There is a natural intention to reduce the time-stepping error as it has proven to be a substantial part of the total forecast error. Yet a new scheme must be non-intrusive, i.e., easily implementable in legacy codes in order to avoid significant programming undertaking. The object of this work is the development, analysis and validation of novel Robert-Asselin type time filters, addressing both of the above problems. Specifically, we first propose and analyze a higher-order Robert-Asselin (hoRA) type time filter. The analysis reveals that the filtered leapfrog scheme exhibits second- or third-order accuracy depending on the filter parameter. We then investigate its behavior when used in conjunction with the implicit-explicit integration, which is commonly used in weather and climate models to relieve the severe time step restriction induced by unimportant high-frequency waves. Next, we build a framework of constructing a family of hoRA filters with any pre-determined order of accuracy. In particular, we focus on the fourth-order time filter. Finally, we present supplemental analysis for several filters developed by Williams. For each direction, we present comprehensive error and stability analysis, and perform numerical tests to verify theoretical results

Turbulence: Numerical Analysis, Modelling and Simulation

The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is available for calibration, the importance of a sound mathematical foundation that addresses the needs of practical computing increases. This Special Issue is directed at this crossroads of rigorous numerical analysis, the physics of turbulence and the practical needs of turbulent flow simulations. It seeks papers providing a broad understanding of the status of the problem considered and open problems that comprise further steps

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com