Adding KP-I example to the example folder for MOLE

[tcalderon-space]

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• Refactor
• Feature
• Bug Fix
• Optimization
• [x ] Example
• Documentation

Description

Adding the example for KP-I (Kadomtsev–Petviashvili equation) in matlab.

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[mdumett]

Kadomtsev-Petviashivili equation (KP) has exact solutions under different set of parameters. It also includes a spatial derivative of the time derivative term that complicates its numerical solution.

This code needs to be tested better before including it in MOLE.

For numerical accuracy and expected behavior to compare to https://arxiv.org/pdf/1605.03213, where one can find appropriates domain size, dx, dt, and time derivative schemes. Another paper where one could find relevant information for the numerical solution of KP is https://www.sciencedirect.com/science/article/pii/S0096300313004633.

In addition, the code requires several improvements. For example, line 5 refers to a non-existing directory in the MOLE repository. Line 22 sets up dt = dx, but give no reason for that. Both dx and dt seem to be large. In line 28, the total number of time steps is set to one. The matrix in line 80 does not depend on the iteration and should be before the time loop. Boundary conditions require special care. In some cases it is known that exact solutions develop lumps.

[valeriabarra]

Hi @tcalderon-space , thank you so much for considering contributing to MOLE!

Perhaps here you meant to use this line for testing/debugging purposes, using only 1 time step? Thank you!

[tcalderon-space]

Kadomtsev-Petviashivili equation (KP) has exact solutions under different set of parameters. It also includes a spatial derivative of the time derivative term that complicates its numerical solution.

This code needs to be tested better before including it in MOLE.

For numerical accuracy and expected behavior to compare to https://arxiv.org/pdf/1605.03213, where one can find appropriates domain size, dx, dt, and time derivative schemes. Another paper where one could find relevant information for the numerical solution of KP is https://www.sciencedirect.com/science/article/pii/S0096300313004633.

In addition, the code requires several improvements. For example, line 5 refers to a non-existing directory in the MOLE repository. Line 22 sets up dt = dx, but give no reason for that. Both dx and dt seem to be large. In line 28, the total number of time steps is set to one. The matrix in line 80 does not depend on the iteration and should be before the time loop. Boundary conditions require special care. In some cases it is known that exact solutions develop lumps.

Thank you for responding! I don't have access to the second article (even while using the school's account), but I will follow the values given in the first article. I did have a question though, if I follow the Zaitsev traveling wave solution (pg 6) the results don't seem to be the same as the one from this page http://scholarpedia.org/article/Kadomtsev-Petviashvili_equation#:~:text=Line%20solitonsThe%20one-soliton%20solution%20of. Under "Line Solitons" will this change anything? When I use the new domain values, my results end up stiffer..

[mdumett]

KP-I has different analytical solutions depending on the parameter values. What was suggested is to provide a similar plots and the tables provided by the authors of that paper.