The Rayleigh-Taylor mixing with a premixed layer is investigated in the present study. A BHR-II turbulence model is employed to simulate the Rayleigh-Taylor mixing with different premixed layers. The implementation of the BHR-II model is described in detail and validated in canonical problems. Afterwards, the classic Rayleigh-Taylor instability (RTI) with a premixed layer is studied. Effects of density ratio, layer thickness and density profiles on the late time behaviour of classic RTI are analysed, and the results are also compared with ILES data. It turns out that the premixed layer delays the transition time of turbulent mixing. The late time behaviours of different premixed cases show a similar trend with the similarity of their distributions of turbulent kinetic energy. It is also shown that the premixed layer has little effect on the tilted angle in the canonical tilted rig case while the temporal evolution of the turbulent mixing at the early beginning is distinctly influenced by the premixed layer.

To simulate the incompressible turbulent flows, two models, known as the simplified and highly stable lattice Boltzmann method (SHSLBM) and large eddy simulation (LES) model, are employed in this paper. The SHSLBM was developed for simulating incompressible viscous flows and showed great performance in numerical stability at high Reynolds numbers, which means that this model is capable of dealing with turbulent flows by adding the turbulence model. Therefore, the LES model is combined with SHSLBM. Inspired by the less amount of grids required for SHSLBM, a local grid refinement method is used at relatively high Reynolds numbers to improve computational efficiency. Several benchmark cases are simulated and the obtained numerical results are compared with the available results in literature, which show excellent agreement together with greater computational performance than other algorithms.

In this paper, a new fifth-order finite volume central weighted essentially non-oscillatory (CWENO) scheme is proposed for solving hyperbolic conservation laws on staggered meshes. The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials (in one dimension) or four linear polynomials (in two dimensions) in a traditional WENO fashion and a time discretization method using the natural continuous extension (NCE) of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme. This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes [37, 46] with the application of smaller number of unequal-sized spatial stencils. Since the new nonlinear weights are adopted, the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in $L^1$ and $L^∞$ norms in smooth regions, and control the spurious oscillations near strong shocks or contact discontinuities. The new CWENO scheme has advantages over the classical CWENO schemes [37, 46] on staggered meshes in its simplicity and easy extension to multi-dimensions. Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifth-order finite volume CWENO scheme.

In this work, spatial second order positivity preserving characteristic block-centered finite difference methods are proposed for solving convection dominated diffusion problems. By using a conservative piecewise parabolic interpolation with positive constraint, the temporal first order scheme is shown to conserve mass exactly and preserve the positivity property of solution. Taking advantage of characteristics, there is no strict restriction on time steps. The scheme is extended to temporal second order by using a particular extrapolation along the characteristics. To restore solution positivity, a mass conservative local limiter is introduced and verified to keep second order accuracy. Numerical examples are carried out to demonstrate the performance of proposed methods.

In this paper, we propose, analyze, and test a new fully discrete, efficient, decoupled, stable, and practically second-order time-stepping algorithm for computing MHD ensemble flow averages under uncertainties in the initial conditions and forcing. For each viscosity and magnetic diffusivity pair, the algorithm picks the largest possible parameter $θ ∈ [0,1]$ to avoid the instability that arises due to the presence of some explicit viscous terms. At each time step, the algorithm shares the same system matrix with all $J$ realizations but has different right-hand-side vectors for all $J$ realizations. This saves assembling time and computer memory, allows the reuse of the same preconditioner, and can take the advantage of block linear solvers. For the proposed algorithm, we prove stability and convergence rigorously. To illustrate the predicted convergence rates of our analysis, numerical experiments with manufactured solutions are given on a unit square domain. Finally, we test the scheme on a benchmark channel flow over a step problem and it performs well.

Neutral Delay Differential Equation (NDDE) is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena, especially on their application in biological and physiological processes. A fifth-order two-point hybrid implicit multistep block method (2PIH5) has been formulated in this research for the numerical solution of Neutral Delay Differential Equation (NDDE). A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5. The order, consistency, and zero-stability for 2PIH5 have been illustrated. The analyses of convergence and stability test have been performed and discussed. The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method. Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE. The proposed method is proved to be comparable with the other existing methods. It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.

Radial Basis Function (RBF) kernels are key functional forms for advanced solutions of higher-order partial differential equations (PDEs). In the present study, a hybrid kernel was developed for meshless solutions of PDEs widely seen in several engineering problems. This kernel, Power-Generalized Multiquadric — Power-GMQ, was built up by vanishing the dependence of $\epsilon$, which is significant since its selection induces severe problems regarding numerical instabilities and convergence issues. Another drawback of $\epsilon$-dependency is that the optimal $\epsilon$ value does not exist in perpetuity. We present the Power-GMQ kernel which combines the advantages of Radial Power and Generalized Multiquadric RBFs in a generic formulation. Power-GMQ RBF was tested in higher-order PDEs with particular boundary conditions and different domains. RBF-Finite Difference (RBF-FD) discretization was also implemented to investigate the characteristics of the proposed RBF. Numerical results revealed that our proposed kernel makes similar or better estimations as against to the Gaussian and Multiquadric kernels with a mild increase in computational cost. Gauss-QR method may achieve better accuracy in some cases with considerably higher computational cost. By using Power-GMQ RBF, the dependency of solution on $\epsilon$ was also substantially relaxed and consistent error behavior were obtained regardless of the selected $\epsilon$ accompanied.

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors. In this paper, a well-balanced weighted compact nonlinear scheme (WCNS) is proposed for shallow water equations in pre-balanced forms. The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term. Some numerical examples in one- and two-dimensions are also presented to demonstrate the well-balanced property, high order accuracy and good shock capturing capability of the proposed scheme.

In this paper, a novel unconditionally energy stable Smoothed Particle Hydrodynamics (SPH) method is proposed and implemented for incompressible fluid flows. In this method, we apply operator splitting to break the momentum equation into equations involving the non-pressure term and pressure term separately. The idea behind the splitting is to simplify the calculation while still maintaining energy stability, and the resulted algorithm, a type of improved pressure correction scheme, is both efficient and energy stable. We show in detail that energy stability is preserved at each full-time step, ensuring unconditionally numerical stability. Numerical examples are presented and compared to the analytical solutions, suggesting that the proposed method has better accuracy and stability. Moreover, we observe that if we are interested in steady-state solutions only, our method has good performance as it can achieve the steady-state solutions rapidly and accurately.