On the Partial Difference Equations of Mathematical Physics

Courant, Richard; Friedrichs, K. O.; Lewy, Hans

doi:10.1007/978-1-4612-2080-0_6

Cited by 507 publications

(429 citation statements)

References 2 publications

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“…More in detail, along with the design of a set of media according to specifications in (15), the source has been modeled within a 2D environment as a current wire excited by a differentiated gaussian pulse of central frequency of 2 GHz, at distance d 1 = 0.35 m from I 1 ; the backscattered signal is then sensed by an ideal probe that measures the e.m. field at the desired lattice point. The computational volume has been discretized in 1 × 1 mm cells, which means that, for the CourantFriedrichs-Lewy condition [20], the time increment dt is bound to the value of 2.357 · 10 −12 s. The number of iterations has been set to 3200, for an overall simulated time of 7.5 ns. The signal to be processed is represented in Fig.…”

Section: Resultsmentioning

confidence: 99%

Towards the Detection of Multiple Reflections in Time-Domain Em Inverse Scattering of Multi-Layered Media

Caorsi¹,

Stasolla²

2012

PIER B

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Abstract-In this paper, a new theoretical approach for the classification of multiple reflections in time-domain e.m. inverse scattering of multi-layered media is presented. The existence of multiples limits the capabilities of inversion algorithms, thus suitable identification and suppression techniques should be applied to reduce this undesired effect. Assuming a scenario composed of loss-less and non-dispersive media, and providing an accurate time delay estimation (TDE) of backscattered signals, the proposed method allows not only to evaluate the presence of multiples and discriminate them from primary reflections, but also to determine their propagation paths. Preliminary tests performed on FDTD simulated data have shown its potentialities to effectively handle multiple reflections and therefore to enhance the e.m. signals backscattered by primary reflectors.

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Section: Resultsmentioning

confidence: 99%

Towards the Detection of Multiple Reflections in Time-Domain Em Inverse Scattering of Multi-Layered Media

Caorsi¹,

Stasolla²

2012

PIER B

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“…Introduce the space discretization ∆x and the time discretization ∆t satisfying the Courant-Friedrichs-Lewy conditions (see [17]) as in [16]. Define ν = ∆t/∆x, and, for all j ∈ Z and all n ∈ N, the points x j+1/2 = j∆x, x j = (j − 1/2) ∆x, the time t n = n∆t, and the cell C n j = {t n } × x j−1/2 , x j+1/2 of length ∆x.…”

Section: The Godunov Methodsmentioning

confidence: 99%

The Godunov method for a 2-phase model

Garavello

Marcellini

2017

Communications in Applied and Industrial Mathematics

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“…These methods are classified according to their representation of approximate solutions. We shall mention the four main ones, beginning with the oldest [55].…”

Section: Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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On the Partial Difference Equations of Mathematical Physics

Cited by 507 publications

References 2 publications

Towards the Detection of Multiple Reflections in Time-Domain Em Inverse Scattering of Multi-Layered Media

Towards the Detection of Multiple Reflections in Time-Domain Em Inverse Scattering of Multi-Layered Media

The Godunov method for a 2-phase model

A review of numerical methods for nonlinear partial differential equations

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