Well Balanced High Order Numerical Methods for Nonlinear Hyperbolic Systems: Application to Design of Mathematical Tools for Early Warning of Natural Disasters

PI: Manuel J. Castro Díaz
co-PI: Carlos Parés Madroñal

Abstract: This project is the natural extension to three years of execution of the so-called “Well-balanced high-order numerical methods for nonlinear hyperbolic systems. Applications to the simulation of geophysical flows ”that has been requested in the recent call of the University of Malaga (ERDF operational program). The main objective of both projects is the development and improvement of numerical models that constitute useful tools in the prediction and management of disasters caused by geophysical flows, such as floods, tsunamis or “storm surges”. Stratified fluids with variable density will also be studied. New models will be developed that incorporate non-hydrostatic effects, and that can be formulated as nonlinear hyperbolic systems. The development of new high-order numerical schemes for these models using DG-ADER-type schemes or generalized Lax-Wendroff-type schemes will be addressed. High-order, well-balanced numerical schemes for balance laws will be developed, analyzing in detail the presence of discontinuous source terms. Numerical schemes will be developed for problems with degenerate diffusion that appear in multispecies kinetic models related to the deposition of droplets or colloidal substances, as well as in long-time scale sedimentation models. The development of semi-implicit numerical schemes with good well-balanced properties for the simulation of problems related to the dragging and suspension of sediments will also be addressed. Finally, the design of new approximate two-dimensional Riemann solvers will be addressed, which are also of interest in the field of computational astrophysics.

From the computing point of view, we will work on the efficient implementation of the models in multi-GPU architectures and their incorporation into the HySEA software developed by our research group. An algorithm based on “machine learning” techniques will also be proposed to improve the quality of the results obtained on coarse meshes. The simulators obtained will be the subject of technology transfer and interdisciplinary scientific collaborations, especially related to the simulation of geophysical flows or astrophysics. In addition, the improvement or development of new models, the numerical methods to solve them and the techniques for their efficient implementation, will also represent a scientific advance in the scope of the project that may be useful for researchers working in areas in which the equations to be solved are similar.