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Volume 10, Issue 1
The Half-Planes Problem for the Level Set Equation

S. Clain & M. Djenno Ngomanda

Int. J. Numer. Anal. Mod., 10 (2013), pp. 99-115.

Published online: 2013-10

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  • Abstract

The paper is dedicated to the construction of an analytic solution for the level set equation in $R^2$ with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We first rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid.

  • Keywords

Riemann problem, Cauchy problem, level set equation, analytical solution, half-planes problem.

  • AMS Subject Headings

65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-99, author = {Clain , S. and Djenno Ngomanda , M.}, title = {The Half-Planes Problem for the Level Set Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {1}, pages = {99--115}, abstract = {

The paper is dedicated to the construction of an analytic solution for the level set equation in $R^2$ with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We first rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/560.html} }
TY - JOUR T1 - The Half-Planes Problem for the Level Set Equation AU - Clain , S. AU - Djenno Ngomanda , M. JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 99 EP - 115 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/560.html KW - Riemann problem, Cauchy problem, level set equation, analytical solution, half-planes problem. AB -

The paper is dedicated to the construction of an analytic solution for the level set equation in $R^2$ with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We first rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid.

Clain , S. and Djenno Ngomanda , M.. (2013). The Half-Planes Problem for the Level Set Equation. International Journal of Numerical Analysis and Modeling. 10 (1). 99-115. doi:
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