James S. Sochacki

Absorbing boundary conditions and surface waves

One of the major problems in numerically simulating waves traveling in the Earth is that an artificial boundary must be introduced to produce unique solutions. To eliminate the spurious reflections introduced by this artificial boundary, we use a damping expression based on analogies to shock absorbers. This method can reduce the amplitude of the reflected wave to any pre‐specified value and is successful for waves at any angle of incidence. The method can eliminate unwanted reflections from the surface, reflections at the corners of the model, and waves reflected off an interface that strike the artificial boundary. Many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary. Stability problems often occur with these boundary conditions. The method we use at the artificial boundary allows use of stable Dirichlet or von Neumann condition...

Interface conditions for acoustic and elastic wave propagation

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

On Taylor-series expansions of residual stress

Although subgrid-scale models of similarity type are insufficiently dissipative for practical applications to large-eddy simulation, in recently published a priori analyses, they perform remarkably well in the sense of correlating highly against exact residual stresses. Here, Taylor-series expansions of residual stress are exploited to explain the observed behavior and “success” of similarity models. Specifically, the first few terms of the exact residual stress τkl are obtained in (general) terms of the Taylor coefficients of the grid filter. Also, by expansion of the test filter, a similar expression results for the resolved turbulent stress tensor Lkl in terms of the Taylor coefficients of both the grid and test filters. Comparison of the expansions for τkl and Lkl yields the grid- and test-filter dependent value of the constant cL in the scale-similarity model of Liu et al. [J. Fluid Mech. 275, 83 (1994)]. Until recently, little attention has been given to issues related to the convergence of such exp...

Finite-difference schemes for elastic waves based on the integration approach

The authors present a second order explicit finite‐difference scheme for elastic waves in 2-D nonhomogeneous media. These schemes are based on integrating the equations of motion and the stress‐free surface conditions across the discontinuities before discretizing them on a grid. As an alternative for the free‐surface treatment, a scheme using zero density above the surface is suggested. This scheme is first order and is shown to be a natural consequence of the integrated equations of motion and is called a vacuum formalism. These schemes remove instabilities encountered in earlier integration schemes. The consistency study reveals a close link between the vacuum formalism and the integrated/ discretized stress‐free condition, giving priority to the vacuum formalism when a material discontinuity reaches the free surface. The two presented free‐surface treatments coincide in the sense of the limit (grid size → 0) for lateral homogeneity at or near the free surface.

A Picard-Maclaurin theorem for initial value PDEs

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on ℜn with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset of ℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

The trade-offs between alternative finite difference techniques used to price derivative securities

Approximating partial differential equations (PDEs) using the finite difference method (FDM) is a common occurrence in contingent claim asset pricing. Past studies have stated that an explicit scheme is being used when a mixed (a combination of an explicit and implicit) scheme is actually being implemented. This study explores the financial and mathematical consequences of using this commonly found method with other FDM approaches. The consequences on valuation and risk management are also briefly addressed.

Forced Differential Equations: Problems to Impact Intuition

Abstract How should our students think about external forcing in differential equations setting, and how can we help them gain intuition? To address this question, we share a variety of problems and projects that explore the dynamics of the undamped forced spring–mass system. We provide a sequence of discovery-based exercises that foster physical and mathematical intuition about polynomial forcing, as we build tools and techniques (including Greens function) to explore the amazing behavior of We encourage the insightful use of a Computer Algebra System, and provide a paired supplemental Maple worksheet.

Connections Between Power Series Methods and Automatic Differentiation

There is a large overlap in the work of the Automatic Differentiation community and those whose use Power Series Methods. Automatic Differentiation is predominately applied to problems involving differentiation, and Power series began as a tool in the ODE setting. Three examples are presented to highlight this overlap, and several interesting results are presented.

A more accurate finite difference approach to the pricing of contingent claims

Numerical approximations of contingent claim partial differential equations (PDEs) are quickly becoming one of the most accepted techniques used in derivative security valuation. The most common methodology is the finite difference method (FDM). This procedure can be used as long as a well-posed PDE can be derived and therefore lends itself to contingent claims. The FDM requires prescribed conditions at the boundary. These boundary conditions are not readily available (at all boundaries) for most contingent claim PDEs. This study presents an accurate and computationally inexpensive method for providing these boundary conditions. These absorbing and adjusting boundary conditions when applied to the contingent claim PDEs presented in this study increased the accuracy of the FDM solution at relatively little cost.