Semi-lagrangian Method For Solving Level Set Extension Equation

In this paper, we study semi-Lagrangian(S-L) methods for solving a Hamilton-Jacobian equation, which extends an interface quantity into a neighborhood ofthe interface. A level set function is used to represent the interface. One S-Lmethod is the frst order method, the other is the formally second order method.The frst order method uses the frst order Euler method for locating the de-parture point, and the bilinear interpolation for the evaluation of the extendedquantity at the departure point. The formally second order method uses a secondorder Runge-Kutta scheme to locate the departure point, together with the third-order ENO scheme for the interpolation. Numerical examples are presented. It isobserved that the frst order S-L method performs better than the second ordermethod. This is due to the jump discontinuity of the normal velocity across theinterface. Numerical calculations using a classic high order Eulerian method isalso presented for comparison purpose.