General Hopscotch Algorithm for the Numerical Solution of Partial Differential Equations

Gourlay, A. R.; McGuire, George

doi:10.1093/imamat/7.2.216

Cited by 102 publications

(30 citation statements)

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“…Gourlay [10] developed Gordon's idea in a general way for two space dimensional parabolic and elliptic problems and showed that the hopscotch method was infact a Peaceman-Rachford (Peaceman and Rachford, [16]) method with the coefficient matrix split in a rather novel way. Gourlay and McGuire [8] proposed a general class of algorithms for numerical solution of partial differential equations. The structure and properties of the algorithms make them particularly easy to implement.…”

Section: Hopscotch Methodsmentioning

confidence: 99%

Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Maritim¹,

Rotich²,

Bitok³

2018

ams

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The hopscotch scheme is a general method for the solution of second order parabolic and elliptic partial differential equations. It has been shown to be a Peaceman-Rachford alternating direction implicit (ADI) process, Peaceman and Rachford [16]. The method lies somewhere between explicit and implicit. It is fast and accurate in solving partial differential equations although it has seen only limited use by researchers in science and engineering discipline. The method has been used to solve the incompressible Navier-Stokes equations. It is suitable for many fluid flow problems because it is simple to program, faster and gives consistent results in computation per iteration and storage and adaptable to vector or parallel computer architecture. In this research hybrid hopscotch method for solving two-dimensional system of Burgers' equations is developed. A solution algorithm for two-dimensional partial differential equation with mixed boundary condition is first derived. Finally hybrid finite schemes for solving Burgers equations are developed. Hybrid Hopscotch-Crank-Nicholson-Du Fort and Frankel Scheme (HP-CN-DF) proved to be stable because the absolute errors did not change irregularly with a small change in the input data. The schemes were 936

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Section: Hopscotch Methodsmentioning

confidence: 99%

Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Maritim¹,

Rotich²,

Bitok³

2018

ams

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“…Solutions to Eq. 1 were generated in parallel with the red-black Gauss Seidel method 23 . The region of interest was a 2D square L = 5.12 cm on each side which was discretized into a square grid with 512 × 512 sites.…”

Section: Methodsmentioning

confidence: 99%

Topological properties of a self-assembled electrical network via ab initio calculation

Stephenson

Lyon

Hübler

2017

Sci Rep

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Interacting electrical conductors self-assemble to form tree like networks in the presence of applied voltages or currents. Experiments have shown that the degree distribution of the steady state networks are identical over a wide range of network sizes. In this work we develop a new model of the self-assembly process starting from the underlying physical interaction between conductors. In agreement with experimental results we find that for steady state networks, our model predicts that the fraction of endpoints is a constant of 0.252, and the fraction of branch points is 0.237. We find that our model predicts that these scaling properties also hold for the network during the approach to the steady state as well. In addition, we also reproduce the experimental distribution of nodes with a given Strahler number for all steady state networks studied.

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“…One may remark that for a given S 2 the projection of this optimal exercise boundary is varying with time in the same way as the usual single american put exercise boundary does. Finally, An illustration of the relative dierence 32 between the European and American prices is represented in the gure 16.…”

Section: American Options With Two-underlying Assetsmentioning

confidence: 99%

“…Nevertheless, these rules are based on proxies and there is always some30 in the case b 1 = b 2 = r 31. The parameters are b 1 = b 2 = r = 5%, σ 1 = σ 2 = 15%, ρ = 1 2 , T − t 0 = 1 12 and K 0 = K 1 = 1 32. The results are very sensible to the parameters.…”

mentioning

confidence: 91%

Topics on Two-State Option Pricing

et al. 2000

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General Hopscotch Algorithm for the Numerical Solution of Partial Differential Equations

Cited by 102 publications

References 0 publications

Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Topological properties of a self-assembled electrical network via ab initio calculation

Topics on Two-State Option Pricing

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