In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

https://www3.nd.edu/~zxu2/acms60790S13/Shu-WENO-notes.pdf

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

/pdf/verification-and-validation-in-scientific-computing-262w5ti0ia.pdf

Verification and Validation in Scientific Computing

Solving partial differential equations by collocation using radial basis functions

A flux splitting scheme is proposed for the general nonequilibrium flow equations with an aim at removing numerical dissipation of Van-Leer-type flux-vector splittings on a contact discontinuity. The scheme obtained is also recognized as an improved Advection Upwind Splitting Method (AUSM) where a slight numerical overshoot immediately behind the shock is eliminated. The proposed scheme has favorable properties: high-resolution for contact discontinuities; conservation of enthalpy for steady flows; numerical efficiency; applicability to chemically reacting flows. In fact, for a single contact discontinuity, even if it is moving, this scheme gives the numerical flux of the exact solution of the Riemann problem. Various numerical experiments including that of a thermo-chemical nonequilibrium flow were performed, which indicate no oscillation and robustness of the scheme for shock/expansion waves. A cure for carbuncle phenomenon is discussed as well.

A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities(Proceedings of the 12th NAL Symposium on Aircraft Computational Aerodynamics)

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation

This paper reports on the results of a three-year research effort aimed at investigating and exploiting the role of physically motivated asymptotic analysis in the design of numerical methods for singular limit problems in fluid mechanics. Such problems naturally arise, among others, in combustion, magneto-hydrodynamics, and geophysical fluid mechanics. Typically, they are characterized by multiple-space and/or -time scales and by the disturbing fact that standard computational techniques fail entirely, are unacceptably expensive, or both. The challenge here is to construct numerical methods which are robust, uniformly accurate, and efficient through different asymptotic regimes and over a wide range of relevant applications. Summaries of multiple-scales asymptotic analyses for low-Mach-number flows, magneto-hydrodynamics at small Mach and Alfven numbers, and of multiple-scales atmospheric flows are provided. These reveal singular balances between selected terms in the respective governing equations within the considered flow regimes. These singularities give rise to problems of severe stiffness, stability, or to dynamic-range issues in straight-forward numerical discretizations. A formal mathematical framework for the multiple scales asymptotics is then summarized by use of the example of multiple-length-scale single-time-scale asymptotics for low-Mach-number flows. The remainder of the paper focuses on the construction of numerical discretizations for the respective full governing equation systems. These discretizations avoid the pitfalls of singular balances by exploiting the asymptotic results. Importantly, the asymptotics are not used here to derive simplified equation systems, which are then solved numerically. Rather, numerical integration of the full equation sets is aimed at and the asymptotics are used only to construct discretizations that do not deteriorate as a singular limit is approached. One important ingredient of this strategy is the numerical identification of a singular limit regime given a set of discrete numerical state variables. This problem is addressed in an exemplary fashion for multiple-length single-time-scale low-Mach-number flows in one space dimension. The strategy allows a dynamic determination of an instantaneous relevant Mach number, and it can thus be used to drive the appropriate adjustment of the numerical discretizations when the singular limit regime is approached.

Asymptotic adaptive methods for multi-scale problems in fluid mechanics

Summation-by-parts operators for correction procedure via reconstruction

Part 1: General Issues, Historical Background, and Future Perspectives - Geomathematics: Its Role, Its Aim, and Its Potential - Navigation on Sea: Topics in the History of Geomathematics- Gauss and Weber's "Atlas des Erdmagnetismus" (1840) Was Not the First: History of the Geomagnetic Atlases - Part 2: Observational and Measurement Key Technologies- Earth Observation Satellite Missions and Data Access- Satellite-to-Satellite Tracking (Low-Low/High-Low SST)- GOCE: Gravitational Gradiometry in a Satellite- Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation- Part 3: Modeling of the System Earth (Geosphere, Cryosphere, Hydrosphere, Atmosphere, Biosphere, Anthroposphere)- Classical Physical Geodesy - Geodetic Boundary Value Problem - Time-Variable Gravity Field and Global Deformation of the Earth- Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution- Spacetime Modelling of the Earth's Gravity Field by Ellipsoidal Harmonics- Multiresolution Analysis of Hydrology and Satellite Gravitational Data Time Varying Mean Sea Level- Self-Attraction and Loading of Oceanic Masses - Unstructured Meshes in Large-Scale Ocean Modeling- Numerical Methods in Support of Advanced Tsunami Early Warning- Gravitational Viscoelastodynamics- Elastic and Viscoelastic Reaction of the Lithosphere to Loads- Use of Multiscale Methods in Geomathematics- Efficient Modeling of Flow and Transport in Porous Media Using- Multiphysics and Multiscale Approaches- Convection Structures of Binary Fluid Mixtures in Porous Media- Numerical Dynamo Simulations: From Basic Concepts to Realistic Models- Mathematical Properties Relevant to Geomagnetic Field Modeling- Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents- Toroidal - Poloidal Decompositions of Electromagnetic Green's Functions in Geomagnetic Induction- Using B-Spline Expansions for Ionosphere Modeling- The Forward and Adjoint Methods of Global Electromagnetic Induction for CHAMP Magnetic Data- Climate Dynamics- Modern Techniques for Numerical Weather Prediction: A Picture Drawn from Kyrill- Radio Occultation via Satellites- Asymptotic Models for Atmospheric Flows- Stokes Problem, Layer Potentials and Regularizations, Multiscale Applications- On High Reynolds Number Aerodynamics - Separated Flows-Turbulence Theory- Analysis of Forest Fire Spreading Theory- Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation- Model-based Visualization of Instationary Geo-Data with Application to Volcano Ash Data- Modeling of Fluid Transport in Geothermal Research- Fractional Diffusion and Wave Propagation- Modeling Deep Geothermal Reservoirs: Recent Advances and Future Problems- Part 4: Analytic, Algebraic, and Operator Theoretical Methods- Noise Models for Ill-Posed Problems- Sparsity in Inverse Geophysical Problems- Multiparameter Regularization in Downward Continuation of Satellite Data- Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics- Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment- Correlation Modeling of the Gravity Field in Classical Geodesy- Inverse Resistivity Problems in Computational Geoscience- Identification of Current Sources in 3D Electrostatics- Numerical Simulation and Inversion for Geo-Electromagnetic Methods- Transmission Tomography in Seismology- Numerical Algorithms for Non-Smooth Optimization Applicable to Seismic Recovery- Strategies in Adjoint Tomography- Potential-field Estimation using Scalar and Vector Slepian Functions at Satellite Altitude- Multidimensional Seismic Compression by Hybrid Transform with Multiscale Based Coding Tomography: Problems and Multiscale Solutions- RFMP: An Iterative Best Basis Algorithm for Inverse Problems in the Geosciences- Material Behavior: Texture and Anisotropy- Rayleigh Wave Dispersive Properties of a Vector Displacement as a Tool for P- and S-wave Velocities Near Surface Profiling- Dynamic Simulation of Land Use Change and Management Effects on Soil N2O Emissions- Part 5: Statistical and Stochastic Methods- Selected Statistical Methods - Statistical Analysis of Climate Series- Oblique Stochastic Boundary-Value Problem- Geodetic Deformation Analysis with Respect to an Extended Uncertainty Budget)- It's All About Statistics Global Gravity Field Modeling from GOCE and Complementary Data- Mixed Integer Estimation and Validation for Next Generation GNSS- Mixed Integer Linear Models- Part 6: Special Function Systems and Methods- Special Functions in Mathematical Geosciences: An Attempt at a Categorization- Clifford Analysis and Harmonic Polynomials- Splines and Wavelets on Geophysically Relevant Manifolds- Slepian Functions and Their Use in Signal Estimation and Spectral Analysis- Dimension Reduction and Remote Sensing Using Modern Harmonic Analysis- Low Discrepancy Methods- Part 7: Computational and Numerical Methods- Radial Basis Function-generated Finite Differences: A Mesh-free Method for Computational Geosciences- Numerical Integration on the Sphere- Fast Spherical/Harmonic Spline Modeling- Multiscale Approximation- Sparse Solutions of Underdetermined Linear Systems- Nonlinear Methods for Dimensionality Reduction- Part 8: Cartographic, Photogrammetric, Information Systems and Methods- Cartography- Map Projections: Cartographic Information Systems- Modeling Uncertainty of Complex Earth Systems in Metric Space- Geometrical Reference System- Analysis of Data from Multi-Satellite Geospace Missions- Geodetic World Height System Unification- Mathematical Foundations of Photogrammetry- Potential Methods and Geoinformation Systems- Geoinformatics

Handbook of geomathematics

Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method.

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Finite volume methods apply directly to the conservation law form of a differential equation system; and they commonly yield cell average approximations to the unknowns rather than point values. The discrete equations that they generate on a regular mesh look rather like finite difference equations; but they are really much closer to finite element methods, sharing with them a natural formulation on unstructured meshes. The typical projection onto a piecewise constant trial space leads naturally into the theory of optimal recovery to achieve higher than first-order accuracy. They have dominated aerodynamics computation for over forty years, but they have never before been the subject of an Acta Numerica article. We shall therefore survey their early formulations before describing powerful developments in both their theory and practice that have taken place in the last few years.

Thomas Sonar

Papers

Asymptotic adaptive methods for multi-scale problems in fluid mechanics

Summation-by-parts operators for correction procedure via reconstruction

Handbook of geomathematics

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Finite volume methods for hyperbolic conservation laws