Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

In this paper, the inverse spectral problem of Sturm-Liouville operator with boundary conditions and jump conditions dependent on the spectral parameter is investigated. Firstly, the self-adjointness of the problem and the eigenvalue properties are given, then the asymptotic formulas of eigenvalues and eigenfunctions are presented. Finally, the uniqueness theorems of the corresponding inverse problems are given by Weyl function theory and inverse spectral data approach.

SOME PROBLEMS ON JUMP CONDITIONS OF SHOCK WAVES IN 3-DIMENSIONAL SOLIDS

Based on the general conservation laws in continuum mechanics, the Eulerian and Lagrangian descriptions of the jump conditions of shock waves in 3-dimensional solids were presented respectively. The implication of the jump conditions and their relations between each other, particularly the relation between the mass conservation and the displacement continuity, were discussed. Meanwhile the shock wave response curves in 3- dimensional solids, i.e. the Hugoniot curves were analysed, which provide the foundation for studying the coupling effects of shock waves in 3-dimensional solids.

The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

Computational simulations of near-continuum gas flow using Navier-Stokes-Fourier equations with slip and jump conditions based on the modal discontinuous Galerkin method

Blunt-body configurations are the most common geometries adopted for non-lifting re-entry vehicles.Hypersonic re-entry vehicles experience different flow regimes during flight due to drastic changes in atmospheric density.The conventional Navier-Stokes-Fourier equations with no-slip and no-jump boundary conditions may not provide accurate information regarding the aerothermodynamic properties of blunt-bodies in flow regimes away from the continuum.In addition,direct simulation Monte Carlo method requires significant computational resources to analyze the near-continuum flow regime.To overcome these shortcomings,the Navier-Stokes-Fourier equations with slip and jump conditions were numerically solved.A mixed-type modal discontinuous Galerkin method was employed to achieve the appropriate numerical accuracy.The computational simulations were conducted for different blunt-body configurations with varying freestream Mach and Knudsen numbers.The results show that the drag coefficient decreases with an increased Mach number,while the heat flux coefficient increases.On the other hand,both the drag and heat flux coefficients increase with a larger Knudsen number.Moreover,for an Apollo-like blunt-body configuration,as the flow enters into non-continuum regimes,there are considerable losses in the lift-to-drag ratio and stability.

The Fictitious Domain Method with Sharp Interface for Elasticity Systems with General Jump Embedded Boundary Conditions

In framework of the fictitious domain methods with immersed interfaces for the elasticity problem,the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method.The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain.Here,we present a cell-centered finite volume method to discretize the fictitious domain problem.The proposed method is numerically validated for different test cases.This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.

Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface.Second order accuracy for the first derivative is obtained as well.The method is based on the Ghost Fluid Method,making use of ghost points on which the value is defined by suitable interface conditions.The multi-domain formulation is adopted,where the problem is split in two sub-problems and interface conditions will be enforced to close the problem.Interface conditions are relaxed together with the internal equations(following the approach proposed in[10]in the case of smooth coefficients),leading to an iterative method on all the set of grid values(inside points and ghost points).A multigrid approach with a suitable definition of the restriction operator is provided.The restriction of the defect is performed separately for both sub-problems,providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient.Numerical tests will confirm the second order accuracy.Although the method is proposed in one dimension,the extension in higher dimension is currently underway[12]and it will be carried out by combining the discretization of[10]with the multigrid approach of[11]for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.

The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method(IIM)to simulate a two-fluid flow of two immiscible fluids with different density and viscosity.Due to the surface tension and the discontinuous fluid properties,the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids.The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface.We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in[Xu,DCDS,Supplement 2009,pp.838-845].We test our method on some canonical two-fluid flows.The results demonstrate that the method can handle large density and viscosity ratios,is second-order accurate in the infinity norm,and conserves mass inside a closed interface.

Wave Propagation Across Acoustic/Biot’s Media:A Finite-Difference Method

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency Biot’s equations(in the porous medium).Interface conditions are introduced to model various hydraulic contacts between the two media:open pores,sealed pores,and imperfect pores.Well-posedness of the initial-boundary value problem is proven.Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context:a fourth-order ADER scheme with Strang splitting for timemarching;a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory;and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution.Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions,demonstrating the accuracy of the approach.

An Iterative Two-Fluid Pressure Solver Based on the Immersed Interface Method

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm.The iteration is constructed by introducing an unsteady term in the pressure Poisson equation.In each iteration step,a Helmholtz equation is solved on the Cartesian grid using FFT.The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across twofluid interfaces.This solver can also be used to solve Poisson equations on irregular domains.

A NUMERICAL METHOD FOR SOLVING THE ELLIPTIC INTERFACE PROBLEMS WITH MULTI-DOMAINS AND TRIPLE JUNCTION POINTS

Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a chal- lenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface problems with multi-domains and triple junctions. The resulting linear system of equations is positive definite if the matrix coefficients for the elliptic equations in the domains are positive definite. Numerical experiments show that this method is about second order accurate in the L~ norm for piecewise smooth solutions. Corner singularity can be handled in a way such that the accuracy does not degenerate. The triple junction is carefully resolved and it does not need to be placed on the grid, giving our method the potential to treat moving interface problems without regenerating mesh.

A Bilinear Petrov-Galerkin Finite Element Method for Solving Elliptic Equation with Discontinuous Coefficients

In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the cell are discussed.The condition number of the large sparse linear system is studied.Numerical results demonstrate that the method is nearly second order accurate in the L^(∞)norm and L^(2) norm,and is first order accurate in the H^(1) norm.

AN IMPROVED NON-TRADITIONAL FINITE ELEMENT FORMULATION FOR SOLVING THE ELLIPTIC INTERFACE PROBLEMS

We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. All possible situations that the interface cuts the grid are considered. Both Diriehlet and Neumann boundary conditions are discussed. The coefficient matrix data can be given only on the grids, rather than an analytical function. Extensive numerical experiments show that this method is second order accurate in the L∞ norm.

ANumericalMethod for Solving Elasticity Equations with Interfaces

Solving elasticity equationswith interfaces is a challenging problemformost existing methods.Nonetheless,it has wide applications in engineering and science.An accurate and efficient method is desired.In this paper,an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces.The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface.The resulting linear system of equations is shown to be positive definite under certain assumptions.Numerical experiments show that thismethod is second order accurate in the L¥norm for piecewise smooth solutions.More than 1.5th order accuracy is observed for solution with singularity(second derivative blows up)on the sharp-edged interface corner.