Mathematical and Computational Engineering Stream

Adaptive Multilevel Wavelet Methods for Scientific Computing

Faculty: Ratikanta Behera (CDS), Thirupathi Gudi (Math)

Summary:

Mathematical modeling of problems in science and engineering typically involves solving nonlinear partial differential equations. These become challenging for problems with localized structures or sharp transitions and strong multiscale character, where spatial and temporal scales may span several orders of magnitude. Such multiscale problems are encountered in several fields of practical interest, such as astrophysics, material sciences, meteorology, and combustion. If the traditional finite difference approach will consider for the numerical solution of such problems, then one requires a very fine grid to incorporate the steep gradients. The numerical solution for such problems on uniform grids is impractical since high-resolution computations will be needed only in regions where sharp transitions occur. The proposed Ph.D. project will aim to design adaptive multilevel wavelet-based high-order (targeted) essentially nonoscillatory (T)ENO schemes for approximating discontinuous functions without oscillations near the discontinuities, and solving partial differential equations with applications. Further, the project will focus on designing simultaneous space–time adaptive multilevel wavelet method for time-dependent partial differential equations to address the shortcomings of the current numerical methods: (1) the inefficiency of using a single time step for all spatial locations and (2) the lack of control of the global error in time. The Ph.D. student should have a good mathematical and preferably computational backgrounds. Knowledge of numerical solutions of PDEs will be an advantage.

References

  • I. Daubechies, Ten lectures on Wavelets. SIAM (1992)
  • O. V. Vasilyev, C. Bowman: Second generation wavelet collocation method for the solution of partial differential equations. Journal of Computational Physics, 165, 660–693 (2000)
  • R. Maulik, O. San, and R. Behera. An adaptive multilevel wavelet framework for scale-selective WENO reconstruction schemes. International Journal for Numerical Methods in Fluids 87, no. 5: 239-269 (2018).
  • R. Behera, M. Mehra, and N. K-R. Kevlahan. Multilevel approximation of the gradient operator on an adaptive spherical geodesic grid. Advances in Computational Mathematics 41: 663-689 (2015).
  • L. Fu, X. Hu, and N. A. Adams. Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. Journal of Computational Physics, 349 (2017), 97–121.
  • L. Fu, X. Hu, and N. A. Adams. A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws. Journal of Computational Physics, 374 (2018), 724–751.