High-Order Sweeping Methods for the Eikonal Equation

The main goal of this work is to construct a high-order fast sweeping method for the Eikonal equation. The fast sweeping method is an iterative method and in each iterate or sweep it solves one paraxial Eikonal equation. The sweeps will be in the positive and negative direction of each dimension. To get the viscosity solution a point is only updated if its new computed value is smaller than the existing value in that point. For stability reasons a modified paraxial Eikonal equation is used, which contains a cut off parameter. To construct a high-order sweeping method we use WENO schemes for the space derivatives and Runge-Kutta steps in time. We show numerically for some test problems that such a method is high-order accurate in all points where the solution is smooth. For these problems the number of sweeps is also independent of the grid size. This implies that the high-order method has optimal complexity, O(N). We also give a suggestion of how to choose the cut off parameter in the paraxial Eikonal equation. Högre ordningens Sweeping-metoder fö