Since the original DDG method has been introduced by Liu et al. [8] in 2009, a variety of DDG type methods have been proposed and further developed. In this paper, we further investigate and develop a new DDG method with interface correction (DDG (IC)) as the discretization of viscous and heat fluxes for the compressible Navier-Stokes equations on unstructured grids. Compared to the original DDG method, the newly developed DDG (IC) method demonstrates its superior in delivering the optimal order of accuracy under demanding situations. Strategies in extension and application of this newly developed DDG (IC) method for solving the compressible Navier-Stokes equations and special treatments designed for handling boundary viscous fluxes are presented and examined in this work. The performance of the new DDG method with interface correction is carefully evaluated and assessed through a number of typical test cases. Numerical experiments show that the new DDG method with interface correction can achieve the optimal order of accuracy on both uniform structured grids and nonuniform unstructured grids, which clearly indicates its potential for further applications of real engineering practices.

The blood flow model admits the steady state, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which exactly preserves the steady state. In order to maintain the well-balanced property, we propose to reformulate the equation and apply a novel source term approximation. Extensive numerical experiments are carried out to verify the performances of the current scheme such as the maintenance of well-balanced property, the ability to capture the perturbations of such steady state and the genuine high order accuracy for smooth solutions.

The method of fundamental solutions (MFS) and the Collocation Trefftz method have been known as two highly effective boundary-type methods for solving homogeneous equations. Despite many attractive features of these two methods, they also experience different aspects of difficulties. Recent advances in the selection of source location of the MFS and the techniques in reducing the condition number of the Trefftz method have made significant improvement in the performance of these two methods which have been proven to be theoretically equivalent. In this paper we will compare the numerical performance of these two methods under various smoothness of the boundary and boundary conditions.

In this research, motion and deformation of a red blood cell (RBC) in a microchannel with stenosis is investigated by combined Lattice Boltzmann-Immersed Boundary method. The fluid flow occurs due to the pressure difference between the inlet and the outlet of the microchannel. The immersed boundary algorithm guarantees that there is no relative velocity between the RBC and fluid. Therefore, mass transfer along the immersed border does not occur. It can be seen that the healthy RBC has more deformation and passes the stenosis more easily while the sick one passes the stenosis with less deformation and returns to its initial state faster. Increasing the pressure gradient (i.e., increasing Reynolds number) would cause more deformation of the RBC. It is found that a healthy RBC moves faster than a sick one along the microchannel. Blood pressure increases due to the presence of stenosis and low deformable RBCs. It is the reason of many serious diseases including cardiovascular diseases. The results of this paper were compared to the previous valid results and good agreements were observed.

Using the standard mixed Galerkin methods with equal order elements to solve Biot's consolidation problems, the pressure close to the initial time produces large non-physical oscillations. In this paper, we propose a class of fully discrete stabilized methods using equal order elements to reduce the effects of non-physical oscillations. Optimal error estimates for the approximation of displacements and pressure at every time level are obtained, which are valid even close to the initial time. Numerical experiments illustrate and confirm our theoretical analysis.

In this paper, we consider the mixed finite element method (MFEM) of the elasticity problem in two and three dimensions (2D and 3D). We develop a new residual based stabilization method to overcome the inf-sup difficulty, and use Langrange elements to approximate the stress and displacement. The new method is unconditionally stable, and its stability can be obtained directly from Céa's lemma. Optimal error estimates for the $H^1$-norm of the displacement and $H$(div)-norm of the stress can be obtained at the same time. Numerical results show the excellent stability and accuracy of the new method.

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme. The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism. The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENO scheme efficiently. Besides, a scaling technique is used to circumvent the growth of condition numbers as mesh refined. However, weak oscillations still appear when the SWENO scheme deals with complex low density equations. In order to guarantee the maximum-principle-preserving (MPP) property, we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy. A novel procedure is designed to prove this property theoretically. Finally, numerical examples for one- and two-dimensional problems are presented to verify the good performance, maximum principle preserving, essentially non-oscillation and high resolution of the proposed scheme.

Natural convection in a square cavity at high Rayleigh numbers is simulated by multiple relaxation time (MRT) lattice Boltzmann method (LBM) with a separate distribution function to solve the temperature. The Rayleigh numbers examined here range from $Ra=10^3$ to $Ra=10^8$. For Rayleigh numbers below $10^8$, the flow remains stationary and transition occurs beyond $Ra=2×10^8$. Unsteady results at higher Rayleigh numbers ($Ra=10^9$ and $Ra=10^{10}$) are also investigated. To the best of our knowledge, this is the first accurate study which involves the high Rayleigh numbers $Ra=10^9$, $10^{10}$.

This paper investigates the dynamic responses of clamped-clamped functionally graded material circular cylindrical shell at both ends with small initial geometric imperfection and subjected to complex loads. The small initial geometric imperfection of the cylindrical shell is characterized with the shape of hyperbolic function. The effects of radial harmonic excitation combined with thermal loads are considered. The classical theory and von-Karman type nonlinear geometric equation are applied to obtain partial differential equation of the functionally gradient material circular cylindrical shell by the Hamilton's principle. The partial differential dynamic equations are truncated by the Galerkin technique, using the modal expansion with the inclusion of axisymmetric and asymmetric modes. The effective material properties vary in the radial direction following a power-law distribution accordance with the volume fractions. The effects of volume fraction indexes, ratios of thickness-radius and length-radius on the first three dimensionless natural frequencies of the perfect cylindrical shell and its counterpart with imperfection are given. The effects of radial external loads, initial geometric imperfections and volume fraction index on the nonlinear dynamic response of the clamped-clamped FGM circular cylindrical shell are discussed by numerical calculation.

This article investigates the geometrically nonlinear free vibration of piezoelectric-piezomagnetic nanobeams subjected to magneto-electro-thermal loading taking into account size effect using the nonlocal elasticity theory. To this end, the size-dependent nonlinear governing equations of motion and corresponding boundary conditions are derived according to the nonlocal elasticity theory and the first-order shear deformation theory with von Kármán-type of kinematic nonlinearity. The effects of size-dependence, shear deformations, rotary inertia, piezoelectric-piezomagnetic coupling, thermal environment and geometrical nonlinearity are taken into account. The generalized differential quadrature (GDQ) method in conjunction with the numerical Galerkin method, periodic time differential operators and pseudo arc-length continuation method is utilized to compute the nonlinear frequency response of piezoelectric-piezomagnetic nanobeams. The influences of various parameters such as non-dimensional nonlocal parameter, temperature change, initial applied electric voltage, initial applied magnetic potential, length-to-thickness ratio and different boundary conditions on the geometrically nonlinear free vibration characteristics of piezoelectric-piezomagnetic nanobeams are demonstrated by numerical examples. It is illustrated that the hardening spring effect increases with increasing the non-dimensional nonlocal parameter, positive initial applied voltage, negative initial applied magnetic potential, temperature rise and decreases with increasing the negative initial applied voltage, positive initial applied magnetic potential and length-to-thickness ratio.

The static response of two-dimensional horizontal layered piezoelectric bounded domain with side face load was investigated. In this paper, the modified scaled boundary finite element method (SBFEM) is provided as an effective semi-analytical methodology. The method is used to solve the static problem for the layered piezoelectric bounded domain. The scaling line definition extends the SBFEM to be more suitable for analyzing the multilayered piezoelectric bounded domain. It avoids the limitations of original SBFEM in modeling the horizontal layered bounded domain. The modified SBFEM governing equation with piezoelectric medium is derived by introducing Duality variable in the Hamilton system. This derivation technology makes the progress concise. The novel displacement and electric governing equations of the modified SBFEM are given together for the first time. The node forces can be expressed as power exponent function with radial coordinate by introducing the auxiliary variable and using the eigenvalue decomposition. The novel modified SBFEM solution of layered bounded domain with piezoelectric medium is solved. The new power expansion function of layered piezoelectric medium with side face load is proposed. This technology significantly extends the application range of modified SBFEM. The novel treatment of side face load for the layered piezoelectric bounded domain is proposed. Numerical studies are conducted to demonstrate the accuracy of proposed technique in handling with the static problem of layered bounded domain with piezoelectric medium for side face load. The influence of the side face load type and depth is discussed in detail.

In this paper, we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schrödinger equation. The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system and discrete the system by using Fourier pseudospectral method in space and symplectic Euler method in time. After composing two different symplectic Euler methods for the ODEs resulted from semi-discretization in space, we get a new explicit scheme for the target equation which is of second order in space and spectral accuracy in time. The canonical Hamiltonian form of the resulted ODEs is presented and the new derived scheme is proved strictly to be symplectic. The new scheme is totally explicit whereas symplectic scheme is generally implicit or semi-implicit. Linear stability analysis is carried out and a necessary Courant-Friedrichs-Lewy condition is given. The numerical results are reported to test the accuracy and efficiency of the proposed method in long-term computing.