In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

/pdf/strong-stability-preserving-high-order-time-discretization-478mkrwfwr.pdf

Strong Stability-Preserving High-Order Time Discretization Methods

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

/pdf/total-variation-diminishing-runge-kutta-schemes-8j998a8qfs.pdf

Total variation diminishing Runge-Kutta schemes

The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a sum- mary of recent invariant-measure characterizations. Then a new statisti- cal derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbi- ased" way) and random samples are then taken from each of these dis- tributions, the significant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empir- ical contexts and helps justify its recent application to computer design, mathematical modelling and detection of fraud in accounting data.

/pdf/a-statistical-derivation-of-the-significant-digit-law-r7fqztvpcz.pdf

A Statistical Derivation of the Significant-Digit Law

A mathematical model of the factional behavior of Teflon sliding bearings for conditions of interest in base isolation is developed. The calibration of the model is based on extensive experimental data that were presented in an accompanying paper. The model is capable of accounting for: (1) Unidirectional and multidirectional motion at the Teflon‐steel interface; (2) velocity and pressure dependence of the coefficient of sliding friction; and (3) breakaway (or static) friction effects. The model is characterized by four parameters. These are the minimum and maximum values of the sliding coefficient of friction, the ratio of breakaway to sliding coefficient of friction at initiation of sliding and a parameter that describes the variation of the sliding coefficient of friction with velocity. Values of these parameters are presented for sixteen combinations of type of Teflon, bearing pressure and condition (surface roughness) of mating steel surface. Applications of the model in the analysis/design of a slid...

Teflon Bearings in Base Isolation II: Modeling

Form of Control Expression % Abstract Form of Control Expression % others 32.4 ! function-call 3.8 object 15.5 object < integer-constant 2.2 object == object 8.9 object > integer-constant 1.8 ! object 7.4 function-call == object 1.6 function-call 7.4 object > object 1.4 expression 5.7 object != integer-constant 1.3 object != object 4.2 function-call == integer-constant 1.2 object == integer-constant 4.0 object < object 1.1 January 30, 2008 v 1.1 15 6.8.4 Selection statements 1740 selection-statements Fu nc tio n de fin iti on s 1 10 100 1,000 10,000 0 25 50 75 100 × × if • • else switch × × ××××××××××××××××××××××××××××××× ×××××× × ××× × ××××× ×× ×××× × ×× × × × × × × ××× × × ××× ×× × ××× × × × ××× × ×

The New C Standard (Excerpted material) An Economic and Cultural Commentary

Implicit two-step Runge–Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge–Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to $y' = ay$ are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge–Kutta theory where the unique two-stage method of order 4 is not semi-implicit.

Two-step Runge-Kutta methods

Boundedness of Solutions of a Nonlinear Nonautonomous Neutral Delay Equation

This paper deals with an initial value problem of a neutral functional-differential equation with variable time delay The effect of the neutral term on the existence and uniqueness of solutions is investigated A comparison theorem is given and is applied to a functional Riccati equation The asymptotic behaviour of solutions of a linear perturbation of the generalized pantograph equation is studied

On neutral functional-differential equations with variable time delays

This paper studies the propagation of discretization error for discontinuou ordinary and retarded differential equations. Various applications are given including one which extends a fundamental theorem of Henrici concerning round-off error.

Numerical solution of ordinary and retarded differntial equations with discontinuou derivatives

Overflow, Underflow, and Severe Loss of Significance in Floating-Point Addition and Subtraction

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