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In this study, a new cell‐centered Lagrangian discontinuous Galerkin (DG) scheme is presented to simulate gas‐water compressible flows. The two‐phase flows are governed by the compressible Euler equation with ideal and stiffened gas equations of state. We integrate the Lagrangian DG scheme with the exact gas‐water Riemann solver for calculating the numerical fluxes so that the inherent stiff features of gas‐water compressible flows are well addressed. Furthermore, in order to guarantee the positivity of the density and internal energy during the high Mach number calculation, the positivity‐preserving limiter and strong stability preserving temporal integral are incorporated to ensure the numerical stability. Six numerical examples are tested to show the accuracy, robustness and positivity‐preserving property of the present scheme. It can be found that the present scheme can handle challenging numerical cases involving large density ratio (up to 1000) and strong shock. The results obtained with the typical HLLC flux are also given for comparison and the present scheme with the exact Riemann solver shows higher accuracy, particularly in the presence of large density ratio at the gas‐water interface. [ABSTRACT FROM AUTHOR]

In this article, three novel algorithms are developed and successfully applied to investigate the stable solutions of time‐fractional nonlinear telegraph equations. Firstly, we presented the shifted Gegenbauer polynomials through appropriate transformations. The approximation of a function u(||x||, t) is defined via shifted Gegenbauer polynomials (SGPs) and then developed the operational matrices of a positive integer and non‐integer derivatives assisted by multi‐dimensional shifted Gegenbauer polynomials vectors. Linearized fully spectral method (LFSM) is proposed by means of function approximations and developed operational matrices whilst Picard iterative scheme is used to linearize the discussed problem. To introduce a semi‐discrete method (SDM), the temporal derivative is approximated through a central difference scheme whereas spatial derivatives are approximated assisted by novel operational matrices. A fully numerical method (FNM) is proposed by introducing the central difference for temporal and spatial variables. Proposed computational methods transform the time‐fractional nonlinear telegraph equation into a system of linear algebraic equations that are easier to tackle. An inclusive comparative study is witnesses that suggested schemes are effective, accurate and well‐matched to investigate the approximate solutions of the problems udnerstudy. Convergence and error bound of the suggested methods are investigated theoretically whereas stability analysis is performed numerically. Moreover, the developed methods can be used conveniently to examine the numerical solution of other multi‐dimensional highly nonlinear fractional or variable order problems of physical nature. [ABSTRACT FROM AUTHOR]