High Order Non Oscillatory Reconstructions and Deep Learning Approach for Numerical Solutions of Hyperbolic Conservation Laws

Abstract

One of the key challenge in numerically approximating solution of hyperbolic conservation laws is newlinethe emergence of spurious oscillations in the presence of discontinuities in the physically correct newlinesolution. Thus development and analysis of numerical methods that achieve high accuracy while newlineremaining free from oscillations have been an active area of research for solving hyperbolic conservation newlinelaws. This thesis initially examines the behavior of non-linear weights in WENO third-order newlineschemes, focusing on their weight distribution and accuracy in view of a smoothness parameter. It is newlinedemonstrated that even for smooth solution region existing prevailing non-linear weights in WENO newlinethird-order schemes tend to deviate from the optimal weights, and thus schemes maintain third-order newlineaccuracy only at specific points along the smoothness parameter spectrum. To address this, the concept newlineof a smooth solution region is presented and utilized to devise a hybrid schemes that combine newlinethe WENO third-order scheme with a upstream third-order scheme. This hybrid approach modifies newlinethe computed non-linear weights into the optimal weights in smooth solution region. The presented newlinenumerical results using third-order hybrid-WENO schemes for various benchmark test problems are newlinegiven and compared with the original third-order WENO schemes with non-modified weights. The newlinecomputational outcomes demonstrate that the proposed hybrid schemes exhibit higher resolution in newlinecapturing solutions with shocks and attain third order convergence rate for smooth solutions. newlineNext, this thesis introduces fifth-order weighted essentially non-oscillatory (WENO) scheme newlineusing new global smoothness indicator which demonstrates improved numerical results compared newlineto the solutions obtained using the fifth order WENO-JS schemes. The proposed scheme achieves newlinean optimal level of approximation, even at critical points where both the first and second derivatives newlinevanishes, but not the third derivative newline