There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

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Table Of Integrals Series And Products

Fractional Differential Equations

Materials with good flexibility and high conductivity that can provide electromagnetic interference (EMI) shielding with minimal thickness are highly desirable, especially if they can be easily processed into films. Two-dimensional metal carbides and nitrides, known as MXenes, combine metallic conductivity and hydrophilic surfaces. Here, we demonstrate the potential of several MXenes and their polymer composites for EMI shielding. A 45-micrometer-thick Ti3C2Tx film exhibited EMI shielding effectiveness of 92 decibels (>50 decibels for a 2.5-micrometer film), which is the highest among synthetic materials of comparable thickness produced to date. This performance originates from the excellent electrical conductivity of Ti3C2Tx films (4600 Siemens per centimeter) and multiple internal reflections from Ti3C2Tx flakes in free-standing films. The mechanical flexibility and easy coating capability offered by MXenes and their composites enable them to shield surfaces of any shape while providing high EMI shielding efficiency.

Electromagnetic interference shielding with 2D transition metal carbides (MXenes)

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online tohelp the readers to develop their own spectral codes for their specific applications.

Spectral Methods: Algorithms, Analysis and Applications

Spectral and High-Order Methods with Applications

Numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy (MBE) require very large time simulation, and therefore large time steps become necessary. The main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit-explicit approach. To this end, an extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Then the stability properties of the resulting schemes are established rigorously. Numerical experiments are carried out to support the theoretical claims. The proposed methods are also applied to simulate the MBE models with large solution times. The power laws for the coarsening process are obtained and are compared with previously published results.

/pdf/stability-analysis-of-large-time-stepping-methods-for-2z6htuknjr.pdf

Stability Analysis of Large Time‐Stepping Methods for Epitaxial Growth Models

We develop efficient moving mesh algorithms for one- and two-dimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and mesh-redistribution. The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula proposed in this work. The iteration for the mesh-redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.

/pdf/adaptive-mesh-methods-for-one-and-two-dimensional-hyperbolic-84j8e45wqz.pdf

Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws

In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.

/pdf/adaptive-finite-element-approximation-for-distributed-57i1cocfgr.pdf

Tao Tang

Papers

Spectral Methods: Algorithms, Analysis and Applications

Spectral and High-Order Methods with Applications

Stability Analysis of Large Time‐Stepping Methods for Epitaxial Growth Models

Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems