Although there are many improvements to WENO3-Z with the target to achieve the optimal order in the occurrence of the first-order critical point ($CP_1$), they mainly address resolution performance, while the robustness of schemes is of less concern and lacks understanding accordingly. In light of our analysis considering the occurrence of critical points within grid intervals, we theoretically prove that it is impossible for a scale-independent scheme that has the stencil of WENO3-Z to fulfill the above order achievement, and current scale-dependent improvements barely fulfill the job when $CP_1$ occurs at the middle of the grid cell. In order to achieve scale-independent improvements, we devise new smoothness indicators that increase the error order from 2 to 4 when $CP_1$ occurs and perform more stably. Meanwhile, we construct a new global smoothness indicator that increases the error order from 4 to 5 similarly, through which new nonlinear weights with regard to WENO3-Z are derived and new scale-independent improvements, namely WENO-${\rm Z}_{ES2}$ and -${\rm Z}_{ES3},$ are acquired. Through 1D scalar and Euler tests, as well as 2D computations, in comparison with typical scale-dependent improvements, the following performances of the proposed schemes are demonstrated: the schemes can achieve third-order accuracy at $CP_1$ no matter its location in the stencil, indicate high resolution in resolving flow subtleties, and manifest strong robustness in hypersonic simulations (e.g., the accomplishment of computations on hypersonic half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well as essentially non-oscillatory solutions of inviscid sharp double cone flow at $M=9.59$), which contrasts the comparative WENO3-Z improvements.

The effects of non-physical mixing on interface development are still not reasonably evaluated when diffuse interface methods (DIMs) are employed to treat the two-medium flows with immiscible interfaces, especially for compressible multi-medium flows with geometrical source terms. In this work, we simulate radially symmetric multi-medium flows employing the sharp interface methods (SIMs) and DIMs to evaluate their performance such as pressure dislocations, mass conservation, and convergence. The $\gamma$-based model and the five-equation transport model are two common DIMs, which are extended to equations with geometrical source terms combined with discontinuous Galerkin (DG) methods. For the SIMs, we employ the classical modified ghost fluid method (MGFM) and its second-order extension (2nd-MGFM) developed recently. Numerical results exhibit that the 2nd-MGFM is more effective in maintaining the interfacial pressure equilibrium relative to the MGFM. The DIMs can always maintain the pressure continuity naturally and total mass conservation, which is not available for SIMs. Further, under the premise of immiscible interfaces defined artificially, the DIMs cannot provide satisfactory single medium mass conservation, while the SIMs have a smaller mass error. In addition, compared to the other three methods, the 2nd-MGFM has higher confidence for radially symmetric flows, matching the exact solution in sparser grids.

In this work, a new lattice Boltzmann model for a class of viscous wave equation is proposed through the variable transformation, which eliminates the mixed third order partial derivative term of time and space. Some numerical tests are performed to validate the present model, and the results show that the present model has a second-order convergence rate in space.

In this paper, we propose two efficient block preconditioners to solve the mass-conserved Ohta-Kawasaki equation with finite element discretization. We also study the spectral distribution of these two preconditioners, i.e., Schur complement preconditioner and the modified Hermitian and skew-Hermitian splitting (MHSS in short) preconditioner. Besides, Newton method and Picard method are used to address the implicitly nonlinear term. We rigorously analyze the convergence of Newton method. Finally, we offer numerical examples to support the theoretical analysis and indicate the efficiency of the proposed preconditioners for the mass-conserved Ohta-Kawasaki equation

In this paper, we develop a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model for simulating weakly compressible isothermal flows, which demonstrates superior stability and accuracy over existing models such as the regularized lattice Boltzmann (RLB) and two-relaxation-time (TRT) models. In this model, a free relaxation parameter, $\tau_{s,2},$ is employed to relax the regularized non-equilibrium third-order terms. Chapman-Enskog analysis reveals that our model can accurately recover the Navier-Stokes equations. Theoretical analysis and numerical experiments both confirm the model’s ability to eliminate non-physical numerical slip associated with the half-way bounce-back scheme. Our simulations of the double shear layer problem and Taylor-Green vortex flow exhibit pronounced advantages in terms of stability and accuracy, even under super-high Reynolds numbers as high as ${\rm Re}=10^7.$ Additionally, the simulation of creeping flow around a square cylinder showcases the model’s precision in computing ultra-low Reynolds numbers down to ${\rm Re}=10^{−7}.$ This robust capability confirms the proposed model as a highly effective and adaptable tool in computational fluid dynamics.

We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variable is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal $L^2$ convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.

An explicit two-order superconvergent weak Galerkin finite element method is designed and analyzed for the heat equation on triangular and tetrahedral grids. For two-order superconvergent $P_k$ weak Galerkin finite elements, the auxiliary inter-element functions must be $P_{k+1}$ polynomials. In order to achieve the superconvergence, the usual $H^1$-stabilizer must be also eliminated. For time-explicit weak Galerkin method, a time-stabilizer is added, on which the time-derivative of the auxiliary variables can be defined. But for the two-order superconvergent weak Galerkin finite elements, the time-stabilizer must be very weak, an $H^{−2}$-like inner-product instead of an $L^2$-like inner-product. We show the two-order superconvergence for both semi-discrete and fully-discrete schemes. Numerical examples are provided.

The Ripa model consists of the shallow water equations and terms which take the horizontal temperature fluctuations into account. The pollutant transport model describes the transport and diffusion of pollutants in shallow water flows. Both models admit hydrostatic solutions in which the gradient of the flux term is exactly balanced by the source term on the right-hand side. In this paper, we write both models in a unified form and propose a well-balanced fifth-order finite difference alternative weighted essentially non-oscillatory (AWENO) scheme, which allows using arbitrary monotone, Lipschitz continuous and consistent numerical flux. For illustration purposes, the Lax-Friedrichs flux is employed. In order to design a well-balanced AWENO scheme, reformulation of the source term and linearization of the WENO interpolation operator are made. The well-balancedness of the proposed method will be analysed theoretically in this paper. Numerical examples verify the well-balanced property, high-order accuracy and effectiveness of our approach.

In this work, the types of shock wave structure for hydro-elastoplastic model under compression are researched. The emphasis focuses on the theory of shock transition in the presence of elastic-plastic-fluid phase transition. As a result, in addition to the classical three-wave structure, two new shock wave patterns are found with the increase of loading strength. Several numerical tests are presented to verify the existence of the three types of wave structure.

In this work, a genuinely two-dimensional method is proposed for elastic-plastic flow in cylindrically symmetric coordinates. The numerical fluxes are obtained by constructing a genuinely two-dimensional approximate Riemann solver that incorporates consideration of the geometric source term and elastic-plastic transition. The resultant genuine 2D numerical flux combines one-dimensional numerical flux in the central region of the cell edge and two-dimensional flux in the cell vertex region to take wave interaction into account. To deal with the geometry singularity, we establish the equations with removed singularity. For a given stationary state, we prove that the present method is well-balanced. Several numerical tests are presented to verify the performances of the proposed method. The numerical results demonstrate the credibility of the present method.

In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation (BDF) and the finite element method (FEM) to solve the incompressible resistive magnetohydrodynamic (MHD) equations. In the discrete scheme, the pressure variable in the fluid field equation is computed through a Poisson equation, and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system. As a result, the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently. We prove the unique solvability, the unconditional energy stability, and particularly optimal error estimates for the proposed scheme. Numerical results are presented to validate the theory of the scheme.

The phenomenon of capillary rise in a tube is mainly affected by the inertia, capillary force, gravitational force, and viscous force of fluids. It has been observed that certain fluid properties and tube geometries can cause the liquid column to oscillate. In this paper, we first present a general second-order ordinary differential equation that can be used to describe the dynamic process of oscillation in terms of the tube length, tube radius, and contact angle. However, it is difficult to obtain its finding an analytical solution owing to the existence of nonlinear terms. Thus, we adopt a deep neural network (DNN) method to solve this equation for its distinct feature in function-fitting ability. Simulations are conducted to test the performance of the DNN method, and the results are found to be in good agreement with data reported in the literature. Based on a suitably designed and trained DNN, we further perform a comprehensive analysis of the rising height and moving velocity of the fluid during the oscillatory process for different combinations of tube length, tube radius, and contact angle. The results show that the DNN method provides an effective tool for studying the oscillation dynamics of capillary rise.