In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Based on the generalizations of the Funk-Hecke formula and the Rayleigh plan-wave expansion formula, an alternative and succinct derivation of the addition theorem for general tensor field is obtained. This new derivation facilitates the diagonalization of the tensor addition theorem. In order to complete this derivation, we have carried out the evaluation of the generalization of the Gaunt coefficient for tensor fields. Since vector fields (special case of tensor fields) are very useful in practice, we discuss vector multipole fields and vector addition theorem in details. The work is important in multiple scattering and fast algorithms in wave physics.

We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations. A non-oscillatory finite volume method for the relaxation system is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation. Our method demonstrated the capability of accurately capturing soliton wave phenomena.

Hypersonic magneto-fluid-dynamic interaction has been successfully performed as a virtual leading-edge strake and a virtual cowl of a rectangular inlet. In a side-by-side experimental and computational study, the magnitude of the induced compression was found to depend on configuration and electrode placement. To better understand the interacting phenomenon the present investigation is focused on a direct current discharge at the leading edge of a cylindrical inlet for which validating experimental data is available. The present computational result is obtained by solving the magneto-fluid-dynamics equations at the low magnetic Reynolds number limit and using a nonequilibrium weakly ionized gas model based on the drift-diffusion theory. The numerical simulation provides a detailed description of the intriguing physics. After validation with experimental measurements, the computed results further quantify the effectiveness of a magneto-fluid-dynamic compression for a hypersonic cylindrical inlet. A minuscule power input to a direct current surface discharge of 8.14 watts per square centimeter of electrode area produces an additional compression of 6.7 percent for a constant cross-section cylindrical inlet.

Local Discontinuous Galerkin (LDG) schemes in the sense of [5] are a flexible numerical tool to approximate solutions of nonlinear convection problems with complicated dissipative terms. Such terms frequently appear in evolution equations which describe the dynamics of phase changes in e.g. liquid-vapour mixtures or in elastic solids. We report on results for one-dimensional model problems with dissipative terms including third-order and convolution operators. Cell entropy inequalities and L²-stability results are proved for those model problems. As is common in phase transition theory the solution structure sensitively depends on the coupling parameter between viscosity and capillarity. To avoid spurious solutions due to the counteracting effect of artificial dissipation by the numerical flux and the actual dissipation terms we introduce Tadmors' entropy conservative fluxes. Various numerical experiments underline the reliability of our approach and also illustrate interesting and (partly) new phase transition phenomena.

Recently, cavitated flows over underwater submerged bodies have attracted researchers to simulate large scale cavitation. Comparatively Computational Fluid Dynamics (CFD) approaches have been used widely and successfully to model developed cavitation. However, it is still a great challenge to accurately predict cavitated flow phenomena associated with interface capturing, viscous effects, unsteadiness and three-dimensionality. In this study, we consider the preconditioned three-dimensional multiphase Navier-Stokes equations comprised of the mixture density, mixture momentum and constituent volume fraction equations. A dual-time implicit formulation with LU Decomposition is employed to accommodate the inherently unsteady physics. Also, we adopt the Roe flux splitting method to deal with flux discretization in space. Moreover, time-derivative preconditioning is used to ensure well-conditioned eigenvalues of the high density ratio two-phase flow system to achieve computational efficiency. Validation cases include an unsteady 3-D cylindrical headform cavitated flow and an 2-D convergent-divergent nozzle channel cavity-problem.

The steady-state heat conduction in heat conductors with temperature dependent thermal conductivity and mixed boundary conditions involving radiation is investigated using the method of fundamental solutions. Various computational issues related to the method are addressed and numerical results are presented and discussed for problems in two and three dimensions.

This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies, based on a multi-component and multiphase flow model which includes heat/moisture convection and conduction/diffusion as well as phase change. A splitting semi-implicit finite volume method is proposed for solving a set of nonlinear convection-diffusion-reaction equations, in which the calculation of liquid water content absorbed by fiber is decoupled from the rest of the computation. The method maintains the conservation of air, vapor and heat flux (energy). Four types of clothing assemblies are investigated and comparison with experimental measurements are also presented.