Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.     

Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models

Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative. This leads to numerical difficulties, which seem to have discouraged further development of complex behavioral continuum models, which is a significant research need.In this paper, we develop a robust numerical scheme that solves hyperbolic traffic flow models based on their non-conservative form. We develop a fifth-order alternative weighted essentially non-oscillatory (A-WENO) finite-difference scheme based on the path-conservative central-upwind (PCCU) method for several non-equilibrium traffic flow models. In order to treat the non-conservative product terms, we use a path-conservative technique. To this end, we first apply the recently proposed second-order finite-volume PCCU scheme to the traffic flow models, and then extend this scheme to the fifth-order of accuracy via the finite-difference A-WENO framework. The designed schemes are applied to three different traffic flow models and tested on a number of challenging numerical examples. Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher. The proposed scheme in this paper sets the stage for developing more robust and complex continuum traffic flow models with respect to human psychological factors.     

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro [25], in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2×2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.     

A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes

In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.     

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

In this paper, the second in a series, we improve the discretization of the higher spatial derivative terms in a spectral volume (SV) context. The motivation for the above comes from [J. Sci. Comput., 46(2), 314–328], wherein the authors developed a variant of the LDG (Local Discontinuous Galerkin) flux discretization method. This variant (aptly named LDG2), not only displayed higher accuracy than the LDG approach, but also vastly reduced its unsymmetrical nature. In this paper, we adapt the LDG2 formulation for discretizing third derivative terms. A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations. The results of the analysis showed that the LDG2 scheme (i) is stable for 2^nd and 3^rd orders and (ii) generates smaller dissipation and dispersion errors than the LDG formulation for all the orders. The 4^th order LDG2 scheme is however mildly unstable: as the real component of the principal eigen value briefly becomes positive. In order to circumvent the above, a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux. Even a weight of 1.5% for the LDG (i.e., 98.5% for the LDG2) was sufficient to make the scheme stable. This weighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation. Numerical experiments are performed to validate the analysis. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.     

Numerical Study of Singularity Formation in Relativistic Euler Flows

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.     

Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.     

A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$) norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.     

Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

This paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi, SIAM Numer. Anal., 49 (2011), pp. 2057–2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied to, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.     

A Two-Phase Flow Simulation of Discrete-Fractured Media Using Mimetic Finite Difference Method

Various conceptual models exist for numerical simulation of fluid flow in fractured porous media, such as dual-porosity model and equivalent continuum model. As a promising model, the discrete-fracture model has been received more attention in the past decade. It can be used both as a stand-alone tool as well as for the evaluation of effective parameters for the continuum models. Various numerical methods have been applied to the discrete-fracture model, including control volume finite difference, Galerkin and mixed finite element methods. All these methods have inherent limitations in accuracy and applicabilities. In this work, we developed a new numerical scheme for the discrete-fracture model by using mimetic finite difference method. The proposed numerical model is applicable in arbitrary unstructured grid cells with full-tensor permeabilities. The matrix-fracture and fracture-fracture fluxes are calculated based on powerful features of the mimetic finite difference method, while the upstream finite volume scheme is used for the approximation of the saturation equation. Several numerical tests in 2D and 3D are carried out to demonstrate the efficiency and robustness of the proposed numerical model.     

A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive method is based on a linear, decoupled scheme introduced by Shen and Yang [30]. An unconditionally energy stable discrete law for the modified energy is shown for the fully discrete scheme. A superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function, respectively. Based on the proposed space and time discretization error estimators, a time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.     

A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.     

Discrete-Velocity Vector-BGK Models Based Numerical Methods for the Incompressible Navier-Stokes Equations

In this paper, we propose a class of numerical methods based on discrete-velocity vector-BGK models for the incompressible Navier-Stokes equations. By analyzing a splitting method with Maxwell iteration, we show that the usual lattice Boltzmann discretization of the vector-BGK models provides a good numerical scheme. Moreover, we establish the stability of the numerical scheme. The stability and second-order accuracy of the scheme are validated through numerical simulations of the two-dimensional Taylor-Green vortex flows. Further numerical tests are conducted to exhibit some potential advantages of the vector-BGK models, which can be regarded as competitive alternatives of the scalar-BGK models.     

A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. In this paper, we construct a lowest-order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. The scheme is based on Lagrange finite element and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the non-self-adjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed problem which does not fit into the framework of classical theoretical analysis. Instead, we obtain the convergence analysis based on the spectral approximation theory of compact operator. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.     

Construction,Analysis and Application of Coupled Compact Difference Scheme in Computational Acoustics and Fluid Flow Problems

In the present work, a new type of coupled compact difference scheme has been proposed for the solution of computational acoustics and flow problems. The proposed scheme evaluates the first, the second and the fourth derivative terms simultaneously. Derived compact difference scheme has a significant spectral resolution and a physical dispersion relation preserving (DRP) ability over a considerable wavenumber range when a fourth order four stage Runge-Kutta scheme is used for the time integration. Central stencil has been used for the present numerical scheme to evaluate spatial derivative terms. Derived scheme has the capability of adding numerical diffusion adaptively to attenuate spurious high wavenumber oscillations responsible for numerical instabilities. The DRP nature of the proposed scheme across a wider wavenumber range provides accurate results for the model wave equations as well as computational acoustic problems. In addition to the attractive feature of adaptive diffusion, present scheme also helps to control spurious reflections from the domain boundaries and is projected as an alternative to the perfectly matched layer (PML) technique.     

Space-Time Discontinuous Galerkin Method for Maxwell's Equations

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.     

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [J. Comput. Phys., 229 (2010), 1448–1466] and it displays the capability in overcoming difficulties such as the start-up error for a single shock, and the numerical instability of the almost stationary shock. In this paper, we will provide the accuracy study and particularly show the performance in simulating 2-D complex wave configurations formulated with the 2-D Riemann problems for compressible Euler equations. For this purpose, we will first review the GRP scheme briefly when combined with the adaptive moving mesh technique and consider the accuracy of the adaptive GRP scheme via the comparison with the explicit formulae of analytic solutions of planar rarefaction waves, planar shock waves, the collapse problem of a wedge-shaped dam and the spiral formation problem. Then we simulate the full set of wave configurations in the 2-D four-wave Riemann problems for compressible Euler equations [SIAM J. Math. Anal., 21 (1990), 593–630], including the interactions of strong shocks (shock reflections), vortex-vortex and shock-vortex etc. This study combines the theoretical results with the numerical simulations, and thus demonstrates what Ami Harten observed "for computational scientists there are two kinds of truth: the truth that you prove, and the truth you see when you compute" [J. Sci. Comput., 31 (2007), 185–193].     

$H^2$-Conforming Methods and Two-Grid Discretizations for the Elastic Transmission Eigenvalue Problem

The elastic transmission eigenvalue problem has important applications in the inverse elastic scattering theory. Recently, the numerical computation for this problem has attracted the attention of the researchers. In this paper, we propose the $H^2$-conforming methods including the classical $H^2$-conforming finite element method and the spectral element method, and establish the two-grid discretization scheme. Theoretical analysis and numerical experiments show that the methods presented in this paper can efficiently compute real and complex elastic transmission eigenvalues.     

Splitting Finite Difference Methods on Staggered Grids for the Three-Dimensional Time-Dependent Maxwell Equations

In this paper, we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain. We propose a new kind of splitting finite-difference time-domain schemes on a staggered grid, which consists of only two stages for each time step. It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions. Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.     

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique not only reduces the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.