Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations by Xiaobing Feng Ohannes Karakashian & Yulong Xing

Author:Xiaobing Feng, Ohannes Karakashian & Yulong Xing
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

3.3 Local Timestepping (LTS).

From here on, we will restrict our attention to piecewise linear approximations and second-order SSP Runge–Kutta timestepping. The LTS method that we employ is described in [9, 21], it is based on a simple modification of the second order SSP Runge–Kutta method described above, to allow for different timesteps in different regions, and to conserve mass.

To describe the method, consider the 1D (one space dimension plus time) scenario shown in Fig. 1. Here we have highlighted three elements, denoted by K j , , where . Element K i−1 has the smallest timestep, which we label . Element K i has timestep and element K i+1 has the largest timestep . We focus on computing the solution in elements K i−1 and K i , given a global solution at time t n . Let and On K i−1, we take two Euler steps:

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