A Compact High Order Space-Time Method for Conservation Laws

This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.     

The Ultra Weak Variational Formulation Using Bessel Basis Functions

We investigate the ultra weak variational formulation (UWVF) of the 2-D Helmholtz equation using a new choice of basis functions. Traditionally the UWVF basis functions are chosen to be plane waves. Here, we instead use first kind Bessel functions. We compare the performance of the two bases. Moreover, we show that it is possible to use coupled plane wave and Bessel bases in the same mesh. As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.     

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb's problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.     

Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in [25]. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.     

The Error-Minimization-Based Strategy for Moving Mesh Methods

The typical elements in a numerical simulation of fluid flow using moving meshes are a time integration scheme, a rezone method in which a new mesh is defined, and a remapping (conservative interpolation) in which a solution is transferred to the new mesh. The objective of the rezone method is to move the computational mesh to improve the robustness, accuracy and eventually efficiency of the simulation. In this paper, we consider the one-dimensional viscous Burgers' equation and describe a new rezone strategy which minimizes the L₂ norm of error and maintains mesh smoothness. The efficiency of the proposed method is demonstrated with numerical examples.     

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.     

Electromagnetic High Frequency Gyrokinetic Particle-in-Cell Simulation

Using the gyrocenter-gauge kinetic theory, an electromagnetic version of the high frequency gyrokinetic numerical algorithm for particle-in-cell simulation has been developed. The new algorithm, being an alternative to a direct Lorentz-force simulation, offers an efficient way to simulate the dynamics of plasma heating and current drive with radio frequency waves. Gyrokinetic formalism enables separation of gyrocenter and gyrophase motions of a particle in a strong magnetic field. From this point of view, a particle may be viewed as a combination of a slow gyrocenter and a quickly changing Kruskal ring. In this approach, the nonlinear dynamics of high frequency waves is described by the evolution of Kruskal rings based on first principles physics. The efficiency of the algorithm is due to the fact that the simulation particles are advanced along the slow gyrocenter orbits, while the Kruskal rings capture fast gyrophase physics. Moreover, the gyrokinetic formalism allows separation of the cold response from kinetic effects in the current, which allows one to use much smaller number of particles than what is required by a direct Lorentz-force simulation. Also, the new algorithm offers the possibility to have particle refinement together with mesh refinement, when necessary. To illustrate the new algorithm, a simulation of the electromagnetic low-hybrid wave propagating in inhomogeneous magnetic field is presented.     

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.     

Development of a Volume of Fluid Method for Computing Interfacial Incompressible Fluid Flows

This study is aimed to develop a volume of fluid (VOF) method to capture the free surface flow. The incompressible two-phase flow is computed by second-order Adams-Bashforth algorithm with a uniform staggered Cartesian grid arrangement. The tangent of hyperbola for interface capturing (THINC) scheme and weighted linear interface calculation (WLIC) based geometrical reconstruction procedure have been implemented in the operator-splitting method for the VOF method. The proposed VOF method preserves mass well, and the interface normal vector can be easily estimated from the level set (LS) function. The LS function, which is a continuous signed distance function around the interface, is represented by solving the re-initialization equation. Numerical results using the present scheme are compared with experimental data and other numerical results in the Rayleigh-Taylor instability, dam-break flow, travelling solitary wave, Kelvin-Helmholtz instability, rising bubble and merging bubble problems. We also present numerical results in detail between computations made with the proposed VOF method and computations made with the conventional LS method.     

A Mortar Spectral Element Method for Full-Potential Electronic Structure Calculations

In this paper, we propose an efficient mortar spectral element approximation scheme for full-potential electronic structure calculations. As a subsequent work of [24], the paper adopts a similar domain decomposition that the computational domain is first decomposed into a number of cuboid subdomains satisfying each nucleus is located in the center of one cube, in which a small ball element centered at the site of the nucleus is attached, and the remainder of the cube is further partitioned into six curvilinear hexahedrons. Specially designed Sobolev-orthogonal basis is adopted in each ball. Classic conforming spectral element approximations using mapped Jacobi polynomials are implemented on the curvilinear hexahedrons and the cuboid elements without nuclei. A mortar technique is applied to patch the different discretizations. Numerical experiments are carried out to demonstrate the efficiency of our scheme, especially the spectral convergence rates of the ground state approximations. Essentially the algorithm can be extended to general eigenvalue problems with the Coulomb singularities.     

Efficient Computation of Nonlinear Crest Distributions for Irregular Stokes Waves

This paper concerns the computation of nonlinear crest distributions for irregular Stokes waves, and a numerical algorithm based on the Fast Fourier Transform (FFT) technique has been developed for carrying out the nonlinear computations. In order to further improve the computational efficiency, a new Transformed Rayleigh procedure is first proposed as another alternative for computing the nonlinear wave crest height distributions, and the corresponding computer code has also been developed. In the proposed Transformed Rayleigh procedure, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure have been applied to calculating the wave crest distributions of a sea state with a Bretschneider spectrum and a sea state with the surface elevation data measured at the Poseidon platform. It is demonstrated in these two cases that the numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure can offer better predictions than those from using the empirical wave crest distribution models. Meanwhile, it is found that our proposed Transformed Rayleigh procedure can compute nonlinear crest distributions more than 25 times faster than the numerical algorithm based on the FFT technique.     

Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers

This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, both methods can not only capture the diffusion limit, but also exhibit uniform convergence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore, a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.     

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.     

Target-Oriented Inversion of Time-Lapse Seismic Waveform Data

Full waveform inversion of time-lapse seismic data can be used as a means of estimating the reservoir changes due to the production. Since the repeated computations for the monitor surveys lead to a large computational cost, time-lapse full waveform inversion is still considered to be a challenging task. To address this problem, we present an efficient target-oriented inversion scheme for time-lapse seismic data using an integral equation formulation with Gaussian beam based Green's function approach. The proposed time-lapse approach allows one to perform a local inversion within a small region of interest (e.g. a reservoir under production) for the monitor survey. We have verified that the T-matrix approach is indeed naturally target-oriented, which was mentioned by Jakobsen and Ursin [24] and allows one to reduce the computational cost of time-lapse inversion by focusing the inversion on the target-area only. This method is based on a new version of the distorted Born iterative T-matrix inverse scattering method. The Gaussian beam and T-matrix are used in this approach to perform the wavefield computation for the time-lapse inversion in the baseline model from the survey surface to the target region. We have provided target-oriented inversion results of the synthetic time-lapse waveform data, which shows that the proposed scheme reduces the computational cost significantly.     

An Interface-Fitted Finite Element Level Set Method with Application to Solidification and Solvation

A new finite element level set method is developed to simulate the interface motion. The normal velocity of the moving interface can depend on both the local geometry, such as the curvature, and the external force such as that due to the flux from both sides of the interface of a material whose concentration is governed by a diffusion equation. The key idea of the method is to use an interface-fitted finite element mesh. Such an approximation of the interface allows an accurate calculation of the solution to the diffusion equation. The interface-fitted mesh is constructed from a base mesh, a uniform finite element mesh, at each time step to explicitly locate the interface and separate regions defined by the interface. Several new level set techniques are developed in the framework of finite element methods. These include a simple finite element method for approximating the curvature, a new method for the extension of normal velocity, and a finite element least-squares method for the reinitialization of level set functions. Application of the method to the classical solidification problem captures the dendrites. The method is also applied to the molecular solvation to determine optimal solute-solvent interfaces of solvation systems.     

Two-Relaxation-Time Lattice Boltzmann Scheme: About Parametrization,Velocity, Pressure and Mixed Boundary Conditions

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach [16, 18] is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions. We propose second- and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the effective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived permeability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations [5, 46] are improved with respect to this property.     

Free Energy Calculations for DNA Near Surfaces Using an Ellipsoidal Geometry

The change in some thermodynamic quantities such as Gibbs' free energy, entropy and enthalpy of the binding of two DNA strands (forming a double helix), while one is tethered to a surface and are analytically calculated. These particles are submerged in an electrolytic solution; the ionic strength of the media allows the linearized version of the Poisson-Boltzmann equation (from the theory of the double layer interaction) to properly describe the interactions [13]. There is experimental and computational evidence that an ion penetrable ellipsoid is an adequate model for the single strand and the double helix [22–25]. The analytic solution provides simple calculations useful for DNA chip design. The predicted electrostatic effects suggest the feasibility of electronic control and detection of DNA hybridization in the fast growing area of DNA recognition.     

Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the "Impulse method" which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency. This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.     

Numerical Entropy and Adaptivity for Finite Volume Schemes

We propose an a-posteriori error/smoothness indicator for standard semi-discrete finite volume schemes for systems of conservation laws, based on the numerical production of entropy. This idea extends previous work by the first author limited to central finite volume schemes on staggered grids. We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement. We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator. The adaptive scheme uses a single nonuniform grid with a variable timestep. We show how to implement a second order scheme on such a space-time non uniform grid, preserving accuracy and conservation properties. We also give an example of a p-adaptive strategy.