The generic structure of solutions of initial value problems of hyperbolic-elliptic systems, also called mixed systems, of conservation laws is not yet fully understood. One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations, so-called oscillatory waves, which are (in general, measure-valued) solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region. To capture these solutions, usually a fine computational grid is required. In this work, a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type. The hyperbolic-elliptic $2 \times 2$ systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension. In the latter case, varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space, giving rise to different kinds of oscillation waves.

The conventional finite difference (FD) schemes are based on the low order polynomial approximation in a local region. This paper shows that when the polynomial approximation is replaced by the multiquadric (MQ) function approximation in the same region, a new FD method, which is termed as MQ-FD method in this work, can be developed. The paper gives analytical formulas of the MQ-FD method and carries out a performance study for its derivative approximation and solution of Poisson equation and the incompressible Navier-Stokes equations. In addition, the effect of the shape parameter $c$ in MQ on the formulas of the MQ-FD method is analyzed. Derivative approximation in one-dimensional space and Poisson equation in two-dimensional space are taken as model problems to study the accuracy of the MQ-FD method. Furthermore, a lid-driven flow problem in a square cavity is simulated by the MQ-FD method. The obtained results indicate that this method may solve the engineering problem very accurately with a proper choice of the shape parameter $c$.

In this paper, transient and steady natural convection heat transfer in an elliptical annulus has been investigated. The annulus occupies the space between two horizontal concentric tubes of elliptic cross-section. The resulting velocity and thermal fields are predicted at different annulus orientations assuming isothermal surfaces. The full governing equations of mass, momentum and energy are solved numerically using the Fourier Spectral method. The heat convection process between the two tubes depends on Rayleigh number, Prandtl number, angle of inclination of tube axes and the geometry and dimensions of both tubes. The Prandtl number and inner tube axis ratio are fixed at 0.7 and 0.5, respectively. The problem is solved for the two Rayleigh numbers of $10^4$ and $10^5$ considering a ratio between the two major axes up to 3 while the angle of orientation of the minor axes varies from $0^\circ$ to $90^\circ$. The results for local and average Nusselt numbers are obtained and discussed together with the details of both flow and thermal fields. For isothermal heating conditions, the study has shown an optimum value for major axes ratio that minimizes the rate of heat transfer between the two tubes. Another important aspect of this paper is to prove the successful use of the Fourier Spectral Method in solving confined flow and heat convection problems.

In this paper, an immersed boundary algorithm is developed by combining the ghost cell method with adaptive tree Cartesian grid method. Furthermore, the proposed method is successfully used to evaluate various inviscid compressible flow with immersed boundary. The extension to three dimensional cases is also achieved. Numerical examples demonstrate the proposed method is effective.

The solution of boundary value problems (BVP) for fourth order differential equations by their reduction to BVP for second order equations, with the aim to use the available efficient algorithms for the latter ones, attracts attention from many researchers. In this paper, using the technique developed by the authors in recent works we construct iterative method for a problem with complicated mixed boundary conditions for biharmonic equation which is originated from nanofluidic physics. The convergence rate of the method is proved and some numerical experiments are performed for testing its dependence on a parameter appearing in boundary conditions and on the position of the point where a transmission of boundary conditions occurs.

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.