Showing papers in "Journal of Scientific Computing in 2013"
TL;DR: The key ingredients behind this fast solver are recursion, efficient low rank factorization using Chebyshev interpolation, and the Sherman–Morrison–Woodbury formula.
Abstract: This article describes a fast direct solver (i.e., not iterative) for partial hierarchically semi-separable systems. This solver requires a storage of $$\mathcal O (N \log N)$$ O ( N log N ) and has a computational complexity of $$\mathcal O (N \log N)$$ O ( N log N ) arithmetic operations. The numerical benchmarks presented illustrate the method in the context of interpolation using radial basis functions. The key ingredients behind this fast solver are recursion, efficient low rank factorization using Chebyshev interpolation, and the Sherman---Morrison---Woodbury formula. The algorithm and the analysis are worked out in detail. The performance of the algorithm is illustrated for a variety of radial basis functions and target accuracies.
146 citations
TL;DR: The method improves the spatial accuracy order of the weighted and shifted Grünwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949) from 2 to 3.
Abstract: In this paper, a compact difference operator, termed CWSGD, is designed to establish the quasi-compact finite difference schemes for approximating the space fractional diffusion equations in one and two dimensions. The method improves the spatial accuracy order of the weighted and shifted Grunwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949 ) from 2 to 3. The numerical stability and convergence with respect to the discrete L 2 norm are theoretically analyzed. Numerical examples illustrate the effectiveness of the quasi-compact schemes and confirm the theoretical estimations.
142 citations
TL;DR: This work analyzes and compares two recent variational models for image denoising and improves their reconstructions by applying a Bregman iteration strategy and discusses efficient numerical realizations of BRegman iterations and modified versions thereof.
Abstract: In this work we analyze and compare two recent variational models for image denoising and improve their reconstructions by applying a Bregman iteration strategy. One of the standard techniques in image denoising, the ROF-model (cf. Rudin et al. in Physica D 60:259---268, 1992), is well known for recovering sharp edges of a signal or image, but also for producing staircase-like artifacts. In order to overcome these model-dependent deficiencies, total variation modifications that incorporate higher-order derivatives have been proposed (cf. Chambolle and Lions in Numer. Math. 76:167---188, 1997; Bredies et al. in SIAM J. Imaging Sci. 3(3):492---526, 2010). These models reduce staircasing for reasonable parameter choices. However, the combination of derivatives of different order leads to other undesired side effects, which we shall also highlight in several examples.
The goal of this paper is to analyze capabilities and limitations of the different models and to improve their reconstructions in quality by introducing Bregman iterations. Besides general modeling and analysis we discuss efficient numerical realizations of Bregman iterations and modified versions thereof.
135 citations
TL;DR: A semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations, focusing in particular on front propagation models in higher dimensions which are related to control problems.
Abstract: We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d=8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.
127 citations
TL;DR: The numerical experiments show that the proposed mean doubly augmented Lagrangian (MDAL) method is not only more efficient than the method proposed by Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), but can also generate recovered images with even higher quality.
Abstract: Wavelet frame based models for image restoration have been extensively studied for the past decade (Chan et al. in SIAM J. Sci. Comput. 24(4):1408---1432, 2003; Cai et al. in Multiscale Model. Simul. 8(2):337---369, 2009; Elad et al. in Appl. Comput. Harmon. Anal. 19(3):340---358, 2005; Starck et al. in IEEE Trans. Image Process. 14(10):1570---1582, 2005; Shen in Proceedings of the international congress of mathematicians, vol. 4, pp. 2834---2863, 2010; Dong and Shen in IAS lecture notes series, Summer program on "The mathematics of image processing", Park City Mathematics Institute, 2010). The success of wavelet frames in image restoration is mainly due to their capability of sparsely approximating piecewise smooth functions like images. Most of the wavelet frame based models designed in the past are based on the penalization of the ? 1 norm of wavelet frame coefficients, which, under certain conditions, is the right choice, as supported by theories of compressed sensing (Candes et al. in Appl. Comput. Harmon. Anal., 2010; Candes et al. in IEEE Trans. Inf. Theory 52(2):489---509, 2006; Donoho in IEEE Trans. Inf. Theory 52:1289---1306, 2006). However, the assumptions of compressed sensing may not be satisfied in practice (e.g. for image deblurring and CT image reconstruction). Recently in Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), the authors propose to penalize the l 0 "norm" of the wavelet frame coefficients instead, and they have demonstrated significant improvements of their method over some commonly used l 1 minimization models in terms of quality of the recovered images. In this paper, we propose a new algorithm, called the mean doubly augmented Lagrangian (MDAL) method, for l 0 minimizations based on the classical doubly augmented Lagrangian (DAL) method (Rockafellar in Math. Oper. Res. 97---116, 1976). Our numerical experiments show that the proposed MDAL method is not only more efficient than the method proposed by Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), but can also generate recovered images with even higher quality. This study reassures the feasibility of using the l 0 "norm" for image restoration problems.
108 citations
TL;DR: A well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography is constructed.
Abstract: In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography The main new technique is a special reconstruction of the flow variables in wet---dry cells, which is presented in this paper for the one dimensional case We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun Math Sci, 5(1):133---160, 2007) The positivity of the computed water height is ensured following (Bollermann et al, Commun Comput Phys, 10:371---404, 2011): The outgoing fluxes are limited in case of draining cells
105 citations
TL;DR: The simple positivity-preserving limiter is reformulated, and it is proved that the resulting scheme guarantees the positivity of the water depth, as well as well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.
Abstract: The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In "Xing et al. Adv. Water Resourc. 33: 1476---1493, 2010)", the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.
98 citations
TL;DR: In this article, a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in a polygonal space is presented.
Abstract: In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in \(\mathbb{R }^d\). For two-dimensional surfaces embedded in \(\mathbb{R }^3\), these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.
97 citations
TL;DR: A stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem and it is proved that the DG-method admits a unique solution under much weaker conditions.
Abstract: We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $$\mathbb R ^{d}$$ R d , $$d\in \{1,2,3\}$$ d ? { 1 , 2 , 3 } . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $$hp$$ hp -version of the finite element method explicitly in terms of the mesh width $$h$$ h , polynomial degree $$p$$ p and wavenumber $$k$$ k . It is shown that the optimal convergence order estimate is obtained under the conditions that $$kh/\sqrt{p}$$ kh / p is sufficiently small and the polynomial degree $$p$$ p is at least $$O(\log k)$$ O ( log k ) . On regular meshes, the first condition is improved to the requirement that $$kh/p$$ kh / p be sufficiently small.
97 citations
TL;DR: High order well-balanced finite difference WENO schemes are designed, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy.
Abstract: The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper
96 citations
TL;DR: A stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults using an isotropic elastic solid cut by pre-existing fault interfaces, resulting in a provably stable discretization.
Abstract: We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.
The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.
The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.
TL;DR: The proposed CT reconstruction model simultaneously reconstructs a high quality image and its corresponding high resolution measurements in Radon domain and is discovered that using the isotropic wavelet frame regularization proposed in Cai et al. in Multiscale Model is superior than using its anisotropic counterpart.
Abstract: X-ray computed tomography (CT) has been playing an important role in diagnostic of cancer and radiotherapy. However, high imaging dose added to healthy organs during CT scans is a serious clinical concern. Imaging dose in CT scans can be reduced by reducing the number of X-ray projections. In this paper, we consider 2D CT reconstructions using very small number of projections. Some regularization based reconstruction methods have already been proposed in the literature for such task, like the total variation (TV) based reconstruction (Sidky and Pan in Phys. Med. Biol. 53:4777, 2008; Sidky et al. in J. X-Ray Sci. Technol. 14(2):119---139, 2006; Jia et al. in Med. Phys. 37:1757, 2010; Choi et al. in Med. Phys. 37:5113, 2010) and balanced approach with wavelet frame based regularization (Jia et al. in Phys. Med. Biol. 56:3787---3807, 2011). For most of the existing methods, at least 40 projections is usually needed to get a satisfactory reconstruction. In order to keep radiation dose as minimal as possible, while increase the quality of the reconstructed images, one needs to enhance the resolution of the projected image in the Radon domain without increasing the total number of projections. The goal of this paper is to propose a CT reconstruction model with wavelet frame based regularization and Radon domain inpainting. The proposed model simultaneously reconstructs a high quality image and its corresponding high resolution measurements in Radon domain. In addition, we discovered that using the isotropic wavelet frame regularization proposed in Cai et al. (Image restorations: total variation, wavelet frames and beyond, 2011, preprint) is superior than using its anisotropic counterpart. Our proposed model, as well as other models presented in this paper, is solved rather efficiently by split Bregman algorithm (Goldstein and Osher in SIAM J. Imaging Sci. 2(2):323---343, 2009; Cai et al. in Multiscale Model. Simul. 8(2):337---369, 2009). Numerical simulations and comparisons will be presented at the end.
TL;DR: The algorithm of Raessi and Pitsch is modified from a staggered grid method to a collocated grid method and their treatment for the nonlinear terms with the variable density, collocated, pressure projection algorithm developed by Kwatra et al.
Abstract: A coupled level set and moment of fluid method (CLSMOF) is described for computing solutions to incompressible two-phase flows. The local piecewise linear interface reconstruction (the CLSMOF reconstruction) uses information from the level set function, volume of fluid function, and reference centroid, in order to produce a slope and an intercept for the local reconstruction. The level set function is coupled to the volume-of-fluid function and reference centroid by being maintained as the signed distance to the CLSMOF piecewise linear reconstructed interface.
The nonlinear terms in the momentum equations are solved using the sharp interface approach recently developed by Raessi and Pitsch (Annual Research Brief, 2009). We have modified the algorithm of Raessi and Pitsch from a staggered grid method to a collocated grid method and we combine their treatment for the nonlinear terms with the variable density, collocated, pressure projection algorithm developed by Kwatra et al. (J. Comput. Phys. 228:4146---4161, 2009). A collocated grid method makes it convenient for using block structured adaptive mesh refinement (AMR) grids. Many 2D and 3D numerical simulations of bubbles, jets, drops, and waves on a block structured adaptive grid are presented in order to demonstrate the capabilities of our new method.
TL;DR: This paper combines two different WenO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries.
Abstract: The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.
TL;DR: An interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b, is analyzed.
Abstract: This short article analyzes an interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b. The procedure obtains its solution by solving a sequence of unconstrained subproblems of minimizing $J(x)+\frac{1}{2}\|Ax-b^{k}\|_{2}^{2}$ , where b k is iteratively updated. In practice, the subproblem at each iteration is solved at a relatively low accuracy. Let w k denote the error introduced by early stopping a subproblem solver at iteration k. We show that if all w k are sufficiently small so that Bregman iteration enters the optimal face, then while on the optimal face, Bregman iteration enjoys an interesting error-forgetting property: the distance between the current point $\bar{x}^{k}$ and the optimal solution set X ? is bounded by ?w k+1?w k ?, independent of the previous errors w k?1,w k?2,?,w 1. This property partially explains why the Bregman iterative procedure works well for sparse optimization and, in particular, for ? 1-minimization. The error-forgetting property is unique to J(x) that is a piece-wise linear function (also known as a polyhedral function), and the results of this article appear to be new to the literature of the augmented Lagrangian method.
TL;DR: This paper study the new effects of Euler’s elastica as the regularization of active contour and validate them numerically using the augmented Lagrangian method.
Abstract: The active contour segmentation model of Chan and Vese has been widely used and generalized in different contexts in the literature. One possible modification is to employ Euler's elastica as the regularization of active contour. In this paper, we study the new effects of this modification and validate them numerically using the augmented Lagrangian method.
TL;DR: The Runge–Kutta discontinuous Galerkin (RKDG) method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions.
Abstract: In this paper we consider Runge---Kutta discontinuous Galerkin (RKDG) schemes for Vlasov---Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions. The RKDG solver for the Vlasov equation is the main focus, while the electric field is obtained through the classical representation by Green's function for the Poisson equation. A rigorous study of recurrence of the DG methods is presented by Fourier analysis, and the impact of different polynomial spaces and the positivity-preserving limiters on the quality of the solutions is ascertained. Several benchmark test problems, such as Landau damping, the two-stream instability, and the Kinetic Electro static Electron Nonlinear wave, are given.
TL;DR: A hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations and optimal convergence for the velocity gradient and pressure and superconvergence for the Velocity.
Abstract: We propose a hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations. We use same polynomial degree to approximate the velocity, its gradient and the pressure. With a special projection and postprocessing, we obtain optimal convergence for the velocity gradient and pressure and superconvergence for the velocity. Numerical results supporting our theoretical results are provided.
TL;DR: The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis.
Abstract: A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions The $$L1$$ discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis The convergence order is $$\mathcal{O }(\tau ^{3-\alpha }+h^4)$$ in the maximum norm, where $$\tau $$ is the temporal grid size and $$h$$ is the spatial grid size, respectively In addition, a Crank---Nicolson scheme is presented and the corresponding error estimates are also established Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of $$\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)$$ is given Then extension to the case with Robin boundary conditions is also discussed Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank---Nicolson scheme are presented to show the effectiveness of the compact scheme
TL;DR: An accelerated linearized Bregman (ALB) method for solving the basis pursuit and related sparse optimization problems andumerical results on compressed sensing and matrix completion problems demonstrate that the ALB method can be significantly faster than the LB method.
Abstract: In this paper, we propose and analyze an accelerated linearized Bregman (ALB) method for solving the basis pursuit and related sparse optimization problems. This accelerated algorithm is based on the fact that the linearized Bregman (LB) algorithm first proposed by Stanley Osher and his collaborators is equivalent to a gradient descent method applied to a certain dual formulation. We show that the LB method requires O(1/?) iterations to obtain an ?-optimal solution and the ALB algorithm reduces this iteration complexity to $O(1/\sqrt{\epsilon})$ while requiring almost the same computational effort on each iteration. Numerical results on compressed sensing and matrix completion problems are presented that demonstrate that the ALB method can be significantly faster than the LB method.
TL;DR: The resulting ROQ rule is found to converge exponentially with the number of nodes, and a factor of $$\sim $$∼50 savings, without loss of accuracy, is observed in evaluations of inner products when ROQ are used as a downsampling strategy for equidistant samples using the trapezoidal rule.
Abstract: We present an algorithm to generate application-specific, global reduced order quadratures (ROQ) for multiple fast evaluations of weighted inner products between parameterized functions. If a reduced basis or any other projection-based model reduction technique is applied, the dimensionality of integrands is reduced dramatically; however, the cost of approximating the integrands by projection still scales as the size of the original problem. In contrast, using discrete empirical interpolation points as ROQ nodes leads to a computational cost which depends linearly on the dimension of the reduced space. Generation of a reduced basis via a greedy procedure requires a training set, which for products of functions can be very large. Since this direct approach can be impractical in many applications, we propose instead a two-step greedy targeted towards approximation of such products. We present numerical experiments demonstrating the accuracy and the efficiency of the two-step approach. The presented ROQ are expected to display very fast convergence whenever there is regularity with respect to parameter variation. We find that for the particular application here considered, one driven by gravitational wave physics, the two-step approach speeds up the offline computations to build the ROQ by more than two orders of magnitude. Furthermore, the resulting ROQ rule is found to converge exponentially with the number of nodes, and a factor of $$\sim $$ ~ 50 savings, without loss of accuracy, is observed in evaluations of inner products when ROQ are used as a downsampling strategy for equidistant samples using the trapezoidal rule. While the primary focus of this paper is on quadrature rules for inner products of parameterized functions, our method can be easily adapted to integrations of single parameterized functions, and some examples of this type are considered.
TL;DR: Two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method and the theory and the usefulness of the implicit–explicit approaches in presence of local refinements are presented.
Abstract: An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit---explicit approaches in presence of local refinements.
TL;DR: Simulations of turbulent flows with and without shock waves indicate that the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies it is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulence flows with shock waves.
Abstract: This paper presents an optimized low-dissipation monotonicity-preserving (MP-LD) scheme for numerical simulations of high-speed turbulent flows with shock waves. By using the bandwidth dissipation optimization method (BDOM), the linear dissipation of the original MP scheme of Suresh and Huynh (J. Comput. Phys. 136, 83---99, 1997) is significantly reduced in the newly developed MP-LD scheme. Meanwhile, to reduce the nonlinear dissipation and errors, the shock sensor of Ducros et al. (J. Comput. Phys. 152, 517---549, 1999) is adopted to avoid the activation of the MP limiter in regions away from shock waves. Simulations of turbulent flows with and without shock waves indicate that, in comparison with the original MP scheme, the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies the MP-LD scheme is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulent flows with shock waves.
TL;DR: An efficient spectral-element method for computing the transmission eigenvalues in two-dimensional radially stratified media is developed, based on a dimension reduction approach which reduces the problem to a sequence of one-dimensional eigenvalue problems that can be efficiently solved by a spectral- element method.
Abstract: We develop an efficient spectral-element method for computing the transmission eigenvalues in two-dimensional radially stratified media. Our method is based on a dimension reduction approach which reduces the problem to a sequence of one-dimensional eigenvalue problems that can be efficiently solved by a spectral-element method. We provide an error analysis which shows that the convergence rate of the eigenvalues is twice that of the eigenfunctions in energy norm. We present ample numerical results to show that the method convergences exponentially fast for piecewise stratified media, and is very effective, particularly for computing the few smallest eigenvalues.
TL;DR: A class of Hamilton–Jacobi equations is derived which models such motions of a vehicle which travels under certain curvature constraints; it unifies two well-known vehicular models, the Dubins’ and Reeds–Shepp’s cars, and gives further generalizations.
Abstract: We propose a PDE approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton---Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins' and Reeds---Shepp's cars, and gives further generalizations. Numerical methods (finite difference for the Reeds---Shepp's car and semi-Lagrangian for the Dubins' car) are investigated for two-dimensional domains and surfaces.
TL;DR: An efficient multigrid method for finite element discretizations of the Stokes equations on both structured grids and unstructured grids and a distributive Gauss–Seidel relaxation as a smoother is shown to be very efficient and outperforms the popular block preconditioned Krylov subspace methods.
Abstract: A distributive Gauss---Seidel relaxation based on the least squares commutator is devised for the saddle-point systems arising from the discretized Stokes equations. Based on that, an efficient multigrid method is developed for finite element discretizations of the Stokes equations on both structured grids and unstructured grids. On rectangular grids, an auxiliary space multigrid method using one multigrid cycle for the Marker and Cell scheme as auxiliary space correction and least squares commutator distributive Gauss---Seidel relaxation as a smoother is shown to be very efficient and outperforms the popular block preconditioned Krylov subspace methods.
TL;DR: A review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary and (ii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint.
Abstract: We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain's boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain's boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.
TL;DR: This work construct extrapolated multirate discretization methods that allows one to efficiently solve problems that have components with different dynamics and provides highly accurate discretizations for the time integration of multiscale ordinary and partial differential equations.
Abstract: In this paper we construct extrapolated multirate discretization methods that allows one to efficiently solve problems that have components with different dynamics. This approach is suited for the time integration of multiscale ordinary and partial differential equations and provides highly accurate discretizations. We analyze the linear stability properties of the multirate explicit and linearly implicit extrapolated methods. Numerical results with multiscale ODEs illustrate the theoretical findings.
TL;DR: This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation and introduces an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy.
Abstract: The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K=kh?0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.
TL;DR: Stability properties of the spectral element method for advection dominated incompressible flows are investigated and errors arising from insufficient quadrature of the convective term, commonly referred to as ‘aliasing errors’, destroy the skew-symmetric properties ofthe convection operator.
Abstract: We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier---Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as `aliasing errors', destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier---Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier---Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.