Magnus Expansion for the Direct Scattering Transform: High-Order Schemes

2021 ◽  
Author(s):  
R. I. Mullyadzhanov ◽  
A. A. Gelash
Keyword(s):  
High Order ◽  
Magnus Expansion ◽  
High Order Schemes ◽  
Direct Scattering ◽  
Scattering Transform

High-orders methods and high-precision arithmetics make direct scattering transform for the Korteweg-De Vries equation robust.

2021 ◽  
Author(s):  
Aleksandr Gudko ◽  
Andrey Gelash ◽  
Rustam Mullyadzhanov
Keyword(s):  
High Precision ◽  
Vries Equation ◽  
High Order ◽  
Scattering Data ◽  
Computational Domain ◽  
Wave Fields ◽  
Scattering Coefficients ◽  
Direct Scattering ◽  
Scattering Transform ◽  
De Vries

<p>Similar to the theory of direct scattering transform for nonlinear wave fields containing solitons within the focusing one-dimensional nonlinear Schrödinger equation [1], we revisit the theory associated with the Korteweg–De Vries equation. We study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense “uncatchable”. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analyzed using high-precision arithmetics and high-order algorithms based on the Magnus expansion [2, 3] providing accurate information about soliton amplitudes, velocities<span>, positions</span> and intensity of the radiation. This procedure is robust even in the presence of noise opening broad perspectives in analyzing experimental data on propagation of surface waves on shallow water.</p><p>The work is partially funded by Russian Science Foundation grant No 19-79-30075.</p><p>[1] Gelash A., Mullyadzhanov R. Anomalous errors of direct scattering transform // Physical Review E 101 (5), 052206, 2020.</p><p>[2] Mullyadzhanov R., Gelash A. Direct scattering transform of large wave packets // Optics Letters 44 (21), 5298-5301, 2019.</p><p>[3] Gudko A., Gelash A., Mullyadzhanov R. High-order numerical method for scattering data of the Korteweg—De Vries equation // Journal of Physics: Conference Series 1677 (1), 012011, 2020.</p><p> </p><p> </p>

scholarly journals Positivity preserving high order schemes for angiogenesis models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Carpio ◽  
E. Cebrian
Keyword(s):  
Kinetic Equation ◽  
Time Discretization ◽  
Strong Stability ◽  
High Order ◽  
Angiogenic Factor ◽  
Strong Stability Preserving ◽  
Positivity Preserving ◽  
Blood Vessel Formation ◽  
High Order Schemes ◽  
Asymptotic Reduction

Abstract Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions.

scholarly journals Compact high-order schemes for the Euler equations

1988 ◽  
Vol 3 (3) ◽  
pp. 275-288 ◽  
Author(s):  
Saul Abarbanel ◽  
Ajay Kumar
Keyword(s):  
Euler Equations ◽  
High Order ◽  
High Order Schemes

High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method

2012 ◽  
Vol 12 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Thibault Pringuey ◽  
R. Stewart Cant
Keyword(s):  
Level Set ◽  
Three Dimensional ◽  
Unstructured Meshes ◽  
High Order ◽  
Convex Polyhedra ◽  
Two Phase ◽  
High Order Schemes ◽  
Flow Problems ◽  
Mixed Element ◽  
Areas Of Interest

AbstractIn this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

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