Showing papers in "Journal of Scientific Computing in 2011"
TL;DR: The convergence of the general algorithm framework is proved under mild assumptions and the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.
Abstract: In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460---489, 2005) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93---111, 2010; Cai et al., SIAM J. Imag. Sci. 2(1):226---252, 2009, CAM Report 09-28, UCLA, March 2009; Yin, CAAM Report, Rice University, 2009) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ? 1 basis pursuit, TV?L 2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.
426 citations
TL;DR: It has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials.
Abstract: The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.
374 citations
TL;DR: This article proposes a simple extension of the well-known Riemann solver of Osher and Solomon to a certain class of hyperbolic systems in non-conservative form, in particular to shallow-water-type and multi-phase flow models and shows applications of the first order version of the proposed path-conservative Osher-type scheme.
Abstract: We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math Comput 38:339---374, 1982) to a certain class of hyperbolic systems in non-conservative form, in particular to shallow-water-type and multi-phase flow models To this end we apply the formalism of path-conservative schemes introduced by Pares (SIAM J Numer Anal 44:300---321, 2006) and Castro et al (Math Comput 75:1103---1134, 2006) For the sake of generality and simplicity, we suggest to compute the inherent path integral numerically using a Gaussian quadrature rule of sufficient accuracy Published path-conservative schemes to date are based on either the Roe upwind method or on centered approaches In comparison to these, the proposed new path-conservative Osher-type scheme has several advantages First, it does not need an entropy fix, in contrast to Roe-type path-conservative schemes Second, our proposed non-conservative Osher scheme is very simple to implement and nonetheless constitutes a complete Riemann solver in the sense that it attributes a different numerical viscosity to each characteristic field present in the relevant Riemann problem; this is in contrast to centered methods or incomplete Riemann solvers that usually neglect intermediate characteristic fields, hence leading to excessive numerical diffusion Finally, the interface jump term is differentiable with respect to its arguments, which is useful for steady-state computations in implicit schemes We also indicate how to extend the method to general unstructured meshes in multiple space dimensions We show applications of the first order version of the proposed path-conservative Osher-type scheme to the shallow water equations with variable bottom topography and to the two-fluid debris flow model of Pitman & Le Then, we apply the higher-order multi-dimensional version of the method to the Baer---Nunziato model of compressible multi-phase flow We also clearly emphasize the limitations of our approach in a special chapter at the end of this article
209 citations
TL;DR: The ability of a Green–Naghdi model to reproduce strongly nonlinear and dispersive wave propagation and the absence of dry areas allows the treatment of the hyperbolic part of the equations to be improved.
Abstract: We investigate here the ability of a Green---Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green---Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration.
107 citations
TL;DR: In this article, it was shown that the eddy viscosity depends on the invariants of the strain rate tensor of the spatially filtered turbulent flow, and the simplest model was then given for a turbulent channel flow (Re?? =590).
Abstract: Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ?Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity ? e has to depend on the invariants $q = \frac{1}{2}\mathrm{tr}(S^{2})$ and $r= -\frac{1}{3}\mathrm{tr}(S^{3})$ of the (filtered) strain rate tensor S. The simplest model is then given by $
u_{e} = \frac{3}{2}(\Delta/\pi)^{2} |r|/q$ . This model is successfully tested for a turbulent channel flow (Re? ? =590).
101 citations
TL;DR: A new efficient formulation of the local space-time discontinuous Galerkin predictor is derived using a nodal approach whose interpolation points are tensor-products of Gauss–Legendre quadrature points, with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier–Stokes equations with chemical reactions.
Abstract: In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971---4001, 2008) to nonlinear systems of advection---diffusion---reaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss---Legendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss---Legendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank---Nicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection---diffusion---reaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier---Stokes equations with chemical reactions.
99 citations
TL;DR: New error estimates are derived for a hybridizable discontinuous Galerkin scheme applied to the Helmholtz equation and the condition number of the condensed hybridized system is seen to be almost independent of the wave number.
Abstract: Finite element methods for acoustic wave propagation problems at higher frequency result in very large matrices due to the need to resolve the wave. This problem is made worse by discontinuous Galerkin methods that typically have more degrees of freedom than similar conforming methods. However hybridizable discontinuous Galerkin methods offer an attractive alternative because degrees of freedom in each triangle can be cheaply removed from the global computation and the method reduces to solving only for degrees of freedom on the skeleton of the mesh. In this paper we derive new error estimates for a hybridizable discontinuous Galerkin scheme applied to the Helmholtz equation. We also provide extensive numerical results that probe the optimality of these results. An interesting observation is that, after eliminating the internal element degrees of freedom, the condition number of the condensed hybridized system is seen to be almost independent of the wave number.
94 citations
TL;DR: This article introduces a class of domain decomposition preconditioners based on the Schwarz framework, and proves bounds on the condition number of the resulting iteration operators of the hp-version discontinuous Galerkin finite element methods.
Abstract: In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.
81 citations
TL;DR: It is shown that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=\mathcal{O}(h^{\frac{1}{2}})$.
Abstract: In this paper, we investigate a scheme for nonlinear reaction-diffusion equations using the mixed finite element methods. To linearize the mixed method equations, we use the two-grid algorithm. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=\mathcal{O}(h^{\frac{1}{2}})$ . As a result, solving such a large class of nonlinear equations will not much more difficult than the solution of one linearized equation.
79 citations
TL;DR: This paper evaluates the parallel performance and scalability of an unstructured grid Shallow Water Equation (SWE) hurricane storm surge model, which is based on the generalized wave continuity equation continuous Galerkin method within a parallel computational framework based on domain decomposition and the MPI library.
Abstract: This paper evaluates the parallel performance and scalability of an unstructured grid Shallow Water Equation (SWE) hurricane storm surge model. We use the ADCIRC model, which is based on the generalized wave continuity equation continuous Galerkin method, within a parallel computational framework based on domain decomposition and the MPI (Message Passing Interface) library. We measure the performance of the model run implicitly and explicitly on various grids. We analyze the performance as well as accuracy with various spatial and temporal discretizations. We improve the output writing performance by introducing sets of dedicated writer cores. Performance is measured on the Texas Advanced Computing Center Ranger machine. A high resolution 9,314,706 finite element node grid with 1 s time steps can complete a day of real time hurricane storm surge simulation in less than 20 min of computer wall clock time, using 16,384 cores with sets of dedicated writer cores.
79 citations
TL;DR: A posteriori error estimates for H1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells are studied.
Abstract: We study in this paper a posteriori error estimates for H 1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.
TL;DR: In this article, an implicit-explicit (IMEX) time discretization was proposed for the initial value problem of the ES-BGK kinetic equation, where the convection is explicit while the relaxation term is implicit to overcome the stiffness.
Abstract: In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.
TL;DR: The numerical examples in this note clearly demonstrate the importance of utilizingMoving-water well-balanced methods for solutions near a moving-water equilibrium.
Abstract: This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations We concentrate on numerical examples with solutions near a moving-water equilibrium For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used On the other hand, moving-water well-balanced methods perform well in these tests The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium
TL;DR: The main goal is to introduce a technique allowing one to add locally and automatically an ‘optimal’ amount of shear stress to make the flow to remain in the hyperbolicity region, and introduce a predictor/corrector strategy.
Abstract: This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an `optimal' amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.
TL;DR: A new finite volume scheme based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, derived from the solution of a local boundary value problem for the entire equation, including the source term.
Abstract: We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.
TL;DR: A fixed point technique is employed which strongly makes use of the stability property of the linearized problem and its finite element approximations to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation.
Abstract: This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampere equation det?(D 2 u 0)=f?(>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74---98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampere equation, submitted). In this approach, the Monge-Ampere equation is approximated by the fourth order quasilinear equation ??Δ2 u ? +det?D 2 u ? =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampere equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ? of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ?, for the error $u^{\varepsilon}-u^{\varepsilon}_{h}$ . Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ? for the error $u^{0}-u_{h}^{\varepsilon}$ , and numerically examine what is the "best" mesh size h in relation to ? in order to achieve these rates.
TL;DR: High resolution simulations using fifth-order weighted essentially non-oscillatory (WENO) schemes with a third-order TVD Runge-Kutta method to examine the features of turbulent mixing layers find the Bandwidth-optimized WENO scheme with total variation relative limiters is found to be less dissipative than the classical WenO scheme, but prone to exhibit some dispersion errors in relatively coarse meshes.
Abstract: This paper reports high resolution simulations using fifth-order weighted essentially non-oscillatory (WENO) schemes with a third-order TVD Runge-Kutta method to examine the features of turbulent mixing layers. The implementation of high-order WENO schemes for multi-species non-reacting Navier-Stokes (NS) solver has been validated through selective test problems. A comparative study of performance behavior of different WENO schemes has been made on a 2D spatially-evolving mixing layer interacting with oblique shock. The Bandwidth-optimized WENO scheme with total variation relative limiters is found to be less dissipative than the classical WENO scheme, but prone to exhibit some dispersion errors in relatively coarse meshes. Based on its accuracy and minimum dissipation error, the choice of this scheme has been made for the DNS studies of temporally-evolving mixing layers. The results are found in excellent agreement with the previous experimental and DNS data. The effect of density ratio is further investigated, reflecting earlier findings of the mixing growth-rate reduction.
TL;DR: It is proved that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest.
Abstract: This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based on the prescription of a family of paths in the phases space. We also consider path-conservative schemes, that were introduced in Pares (SIAM J. Numer. Anal. 44:300, 2006). The first goal is to prove a Lax-Wendroff type convergence theorem. In Castro et al. (J. Comput. Phys. 227:8107, 2008) it was shown that, for general nonconservative systems a rather strong convergence assumption is needed to prove such a result. Here, we prove that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest. Next, we study the relationship between the well-balanced properties of path-conservative schemes applied to systems of balance laws and the family of paths.
TL;DR: This work proposes and analyzes several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems.
Abstract: We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.
TL;DR: A new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem is presented and it is proved that they are stable and error estimates for both the stress field and the displacement field are obtained.
Abstract: We present a new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem. Based on the theory of mixed finite element methods, we prove that they are stable and obtain error estimates for both the stress field and the displacement field. Using the finite element spaces in this family, an exact sequence is established as a discrete version of the elasticity complex in two dimensions. And the relationship between this discrete version and the original one is shown in a commuting diagram.
TL;DR: This is the first global superconvergence analysis of nonconforming mixed finite elements for the Maxwell’s equations and the approximation accuracy will be improved by extrapolation method.
Abstract: In this paper, a nonconforming mixed finite element approximating to the three-dimensional time-harmonic Maxwell's equations is presented. On a uniform rectangular prism mesh, superclose property is achieved for electric field E and magnetic filed H with the boundary condition E×n=0 by means of the asymptotic expansion. Applying postprocessing operators, a superconvergence result is stated for the discretization error of the postprocessed discrete solution to the solution itself. To our best knowledge, this is the first global superconvergence analysis of nonconforming mixed finite elements for the Maxwell's equations. Furthermore, the approximation accuracy will be improved by extrapolation method.
TL;DR: A two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space and solving the quasi-linear elliptic problem on a coarse space.
Abstract: In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r?1 for a class of quasi-linear elliptic problems in ???2. We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H 1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ? d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in ???3.
TL;DR: A novel algorithm for computing high quality approximations of the solitary waves in 1-, 2- and 3 dimensions is explored and used to benchmark a semi-Lagrangian Crank-Nicolson scheme for a finite element discretization of the time dependent problem.
Abstract: We present a model problem for benchmarking codes that investigate magma migration in the Earth's interior. This system retains the essential features of more sophisticated models, yet has the advantage of possessing solitary wave solutions. The existence of such exact solutions to the nonlinear problem make it an excellent benchmark problem for combinations of solver algorithms. In this work, we explore a novel algorithm for computing high quality approximations of the solitary waves in 1-, 2- and 3 dimensions and use them to benchmark a semi-Lagrangian Crank-Nicolson scheme for a finite element discretization of the time dependent problem.
TL;DR: Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process and this new procedure is also effective for higher order WENO schemes and for WENNO schemes with different smoothness indicators.
Abstract: The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202---228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the "essentially non-oscillatory" property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.
TL;DR: This paper develops a variant of the LDG, which not only retains its attractive features, but also vastly reduces its unsymmetrical nature and has a milder stability constraint than the original LDG formulation.
Abstract: The local discontinuous Galerkin (LDG) viscous flux formulation was originally developed by Cockburn and Shu for the discontinuous Galerkin setting and later extended to the spectral volume setting by Wang and his collaborators. Unlike the penalty formulations like the interior penalty and the BR2 schemes, the LDG formulation requires no length based penalizing terms and is compact. However, computational results using LDG are dependant of the orientation of the faces especially for unstructured and non uniform grids. This results in lower solution accuracy and stiffer stability constraints as shown by Kannan and Wang. In this paper, we develop a variant of the LDG, which not only retains its attractive features, but also vastly reduces its unsymmetrical nature. This variant (aptly named LDG2), displayed higher accuracy than the LDG approach and has a milder stability constraint than the original LDG formulation. In general, the 1D and the 2D numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.
TL;DR: The main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality, and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control Problems.
Abstract: In this paper, we propose a mixed variational scheme for optimal control problems with point-wise state constraints, the main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality. Then mixed form based on this fourth order variational inequality is formulated and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control problems. The a priori and a posteriori error estimates are proved for the mixed finite element scheme. Numerical experiments confirm the efficiency of the new strategy.
TL;DR: It is shown that the determination of LES quality based on one error measure alone, can lead to misleading results, related to underlying error-balancing mechanisms, and this problem can be avoided by considering a range of errors simultaneously, emphasizing different scales in the solution.
Abstract: A review is presented of the error-landscape methodology. This approach evaluates the error-response surface of large-eddy simulations (LES) to essential model and numerical parameters by a systematic variation of these parameters. Using an error landscape constructed for LES of decaying homogeneous isotropic turbulence, it is shown that the determination of LES quality based on one error measure alone, can lead to misleading results, related to underlying error-balancing mechanisms. This problem can be avoided by considering a range of errors simultaneously, emphasizing different scales in the solution. Subsequently, the error-landscape method is further illustrated by comparing different numerical discretizations for Smagorinsky LES. Finally, a more complex case, i.e. a high (infinite) Reynolds number boundary layer, is considered.
TL;DR: The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes.
Abstract: The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427---1472, 2010). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360---373, 1992) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479---483, 1999) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ?-stable, well-balanced, and it doesn't require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.
TL;DR: The purpose of the paper is to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they might relate to LES and to introduce an entropy viscosity technique based on the notion of suitable weak solution.
Abstract: This paper focuses on the notion of suitable weak solutions for the three-dimensional incompressible Navier-Stokes equations and discusses the relevance of this notion to Computational Fluid Dynamics. The purpose of the paper is twofold (i) to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they might relate to LES (ii) to introduce an entropy viscosity technique based on the notion of suitable weak solution and to illustrate numerically this concept.
TL;DR: The f-wave version of the wave-propagation algorithm is one approach, but requires the use of a particular averaged value of the source terms at each cell interface in order to be “well balanced” and exactly maintain steady states.
Abstract: Systems of hyperbolic partial differential equations with source terms (balance laws) arise in many applications where it is important to compute accurate time-dependent solutions modeling small perturbations of equilibrium solutions in which the source terms balance the hyperbolic part. The f-wave version of the wave-propagation algorithm is one approach, but requires the use of a particular averaged value of the source terms at each cell interface in order to be "well balanced" and exactly maintain steady states. A general approach to choosing this average is developed using the theory of path conservative methods. A scalar advection equation with a decay or growth term is introduced as a model problem for numerical experiments.