An L ¥ -L p Space-Time Anisotropic Mesh Adaptation Strategy for Time-Dependent Problems

The prediction of physical multidimensional processes involving discontinuities, when addressed with non mesh-adaptive approximations, shows generally a poor convergence rate to exact solution, this rate being typically limited to first-order or worse. Conversely, a good mesh-adaptive approximation should recover higher order accuracy, and more precisely enjoy the following Early Capturing (EC) property: "a mesh adaptive method enjoys the Early Capturing property if it is able to converge to discontinuous solutions at the same rate as for smooth solutions". This property for second-order flow solver has been proved and illustrated in the case of steady inviscid flows [3]. In this work, the error sensor is the main term of the interpolation error of a typical solution field. The interpolation error is controlled globally in L p -norm, generally L 2 is chosen, which provides a mesh size prescription depending on the Hessian of the selected solution field. A fixed point is applied for solving the non-linear interaction between mesh and sensor. It can be demonstrated that such an adaptive process is second-order convergent when the mesh is adapted for best P1 interpolant. This presentation discusses the contribution of anisotropic mesh adaptation to high-order convergence of unsteady flow simulations on complex geometries. Recovering the theoretical second-order convergence for time-dependent problems relies on four fundamental points: a L p -norm control of the interpolation error, as already applied for steady flows [3]