Current Research Interests

My research interests focus on applied mathematics, specifically numerical analysis, Spectral methods,discontinuous Galerkin methods, shock detection algorithms, and high-order WENO-type nonlinear schemes for hyperbolic conservation laws on high dimensional complex manifolds. I also have experience in computational fluid dynamics, high-performance parallel computing, and using artificial neural networks to solve PDEs.

 

 

Selected Publications

• Borges, R., Carmona, M., Costa, B., and Don, W.S., (2008). An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics 227 (6), 3191-3211.
• Castro, M., Costa, B., and Don, W.S., (2011). High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. Journal of Computational Physics 230 (5), 1766-1792.
• Don, W.S., and Borges, R., (2013). Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. Journal of Computational Physics 250, 347-372.
• Don, W.S., and Gottlieb, D., (1994). The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points. SIAM Journal on Numerical Analysis 31 (6), 1519-1534.
• Costa, B., and Don, W.S., (2007). High-order hybrid central-WENO finite difference scheme for conservation laws. Journal of Computational and Applied Mathematics 204 (2), 209-218.
• Schilling, O., Latini, M., and Don, W.S., (2007). Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability. Physical Review E 76 (2), 026319.
• Jacobs, G.B., and Don, W.S., (2009). A high-order WENO-Z finite difference-based particle-source-in-cell method for computation of particle-laden flows with shocks. Journal of Computational Physics 228 (5), 1365-1379.
• Latini, M., Schilling, O., and Don, W.S., (2007). High-resolution simulations and modeling of reshocked single-mode Richtmyer-Meshkov instability: Comparison to experimental data and to amplitude growth model predictions. Physics of Fluids 19 (2).
• Don, W.S., (1994). Numerical study of pseudospectral methods in shock wave applications. Journal of Computational Physics 110 (1), 103-111.
• Tian, K.B., Don, W.S., and Wang, B.S., (2023). High order WENO finite difference scheme with adaptive dual order ideal weights for hyperbolic conservation laws. Applied Numerical Mathematics 187, 50-70.
• Wang, B.S., Don, W.S., Kurganov, A., and Liu, Y., (2023). Fifth-order A-WENO schemes based on the adaptive diffusion central-upwind Rankine-Hugoniot fluxes. Communications on Applied Mathematics and Computation, 1-20.
• Li, P., Li, T., Don, W.S., and Wang, B.S., (2023). Scale-invariant multi-resolution alternative WENO scheme for the Euler equations. Journal of Scientific Computing 94 (1), 15.
• Wang, Y., Wang, B.S., Ling, L., and Don, W.S., (2022). A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds. Journal of Scientific Computing 93 (3), 84.
• Wang, B.S., and Don, W.S., (2022). Affine-invariant WENO weights and operator. Applied Numerical Mathematics 181, 630-646.
• Don, W.S., Li, R., Wang, B.S., and Wang, Y., (2022). A novel and robust scale-invariant WENO scheme for hyperbolic conservation laws. Journal of Computational Physics 448, 110724.
• Gao, Z., Liu, Q., Hesthaven, J.S., Wang, B.S., Don, W.S., and Wen, X., (2021). Non-intrusive reduced order modeling of convection dominated flows using artificial neural networks with application to Rayleigh-Taylor instability. Commun. Comput. Phys 30, 97-123.