Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients.
We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be …
We investigate the properties of discretizations of advection equations on non-cartesian grids and graphs in general. Advection equations discretized on non-cartesian grids have remained a long-standing challenge as the structure of the grid can lead to strong oscillations in the solution, even for otherwise constant velocity fields. We introduce a new method to track oscillations of the solution for rough velocity fields on any graph. The method in particular highlights some inherent structural conditions on the mesh for propagating regularity on solutions.
A High-Order Modified Finite Volume WENO Method on 3D Cartesian Grids
School of Mathematics and Systems Science, Beihang University（北京航空航天大学数学与系统科学学院）；ICMSEC and LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
The \emph{deterministic} sparse grid method, also known as Smolyak's method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about \emph{randomized} versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the $D$-dimensional unit cube $[0,1)^D$. Let $d,s \in \mathbb{N}$ be such that $D=d\cdot s$. The $s$-dimensional building blocks of the sparse grid quadratures are based on stratifie…
We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys. 252 (2013), pp. 606-624] which uses projections on coordinate planes. Assuming it is given as a level set, the surface is represented by a set of points at which it intersects the intervals between grid points in a three-dimensional grid. They are designated as primary or secondary. Discrete functions on the surface have independent values at primary points, with values at secondary p…
Finite volume schemes for hyperbolic conservation laws require a numerical intercell flux. In one spatial dimension the numerical flux can be successfully obtained by solving (exactly or approximately) Riemann problems that are introduced at cell interfaces. This is more challenging in multiple spatial dimensions. The active flux scheme is a finite volume scheme that considers continuous reconstructions instead. The intercell flux is obtained using additional degrees of freedom distributed along the cell boundary. For their time evolution an exact evolution operator is employed, which naturall…
We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart-Thomas polynomials and the DG scheme is based on evolving certain moments of these polynomials which automatically guarantees divergence-free property. We also develop HLL-type multi-dimensional Riemann solvers to estimate the electric field at vertices which are consistent with the 1-D Riemann solvers. When limiters are used, t…
A new parametric class of semi-implicit numerical schemes for a level set advection equation on Cartesian grids is derived and analyzed. An accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial Lax-Wendroff procedure and numerical von Neumann stability analysis. The obtained semi-implicit kappa-scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one-dimensional scheme that is not the case for analogous fully explicit or fully implicit kappa-schemes. A further impr…
We construct a new nonlinear finite volume (FV) scheme for highly anisotropic diffusion equations, that satisfies the discrete minimum-maximum principle. The construction relies on the linearized scheme satisfying less restrictive monotonicity conditions than those of an M-matrix, based on a weakly regular matrix splitting and using the Cartesian structure of the mesh (extension to quadrilateral meshes is also possible). The resulting scheme, obtained by expressing fluxes as nonlinear combinations of linear fluxes, has a larger stencil than other nonlinear positivity preserving or minimum-maxi…
Finite volume schemes often have difficulties to resolve the low Mach number (incompressible) limit of the Euler equations. Incompressibility is only non-trivial in multiple spatial dimensions. Low Mach fixes, however generally are applied to the one-dimensional method and the method is then used in a dimensionally split way. This often reduces its stability. Here, it is suggested to keep the one-dimensional method as it is, and only to extend it to multiple dimensions in a particular, all-speed way. This strategy is found to lead to much more stable numerical methods. Apart from the conceptua…
An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin condition. We introduce the Reconstruction Off-site Data (ROD) method, that transfers in polynomial functions the information located on the physical boundary. Three major advantages are: (1) a simple description of the physical boundary with Robin condition using a collection of points; (2) no analytical expression (implicit or explicit) is required, particularly the ghost cell centroids' projection are not needed; (3)…
We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with eometrically flexible discontinuous Galerkin methods by using overset grids. Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approach a structured Cartesian Hermite method. Unlike many other overset methods we do not need to add artificial dissipation but we find that the built in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain stabili…
Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and sometimes ill-conditioned operators. We review common discretization methods for elliptic PDEs such as the finite difference, finite volume, finite element, and spectral methods and the linear systems they form. We also provide an overview of classic to modern solution methods for the linear systems formed by these discretization methods. These methods include s…
We construct a new volume preserving map from the unit ball $\mathbb B^3$ to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of $L^2(\mathbb B^3)$, consisting in piecewise constant functions with small local support.
The constant center frequency to bandwidth ratio (Q-factor) of wavelet transforms provides a very natural representation for audio data. However, invertible wavelet transforms have either required non-uniform decimation -- leading to irregular data structures that are cumbersome to work with -- or require excessively high oversampling with unacceptable computational overhead. Here, we present a novel decimation strategy for wavelet transforms that leads to stable representations with oversampling rates close to one and uniform decimation. Specifically, we show that finite implementations of th…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a regular but oversampled grid that is defined on a bounding box. This approach allows the use of high order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approxi…
We present a class of hybrid FD-FV (finite difference and finite volume) methods for solving general hyperbolic conservation laws written in first-order form. The presentation focuses on one- and two-dimensional Cartesian grids; however, the generalization to higher dimensions is straightforward. These methods use both cell-averaged values and nodal values as dependent variables to discretize the governing partial differential equation (PDE) in space, and they are combined with method of lines for integration in time. This framework is absent of any Riemann solvers while it achieves numerical …
We present a wavelet-based adaptive method for computing 3D multiscale flows in complex, time-dependent geometries, implemented on massively parallel computers. While our focus is on simulations of flapping insects, it can be used for other flow problems, including turbulence, as well. The incompressible fluid is modeled with an artificial compressibility approach in order to avoid solving elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differe…
A ghost-cell immersed boundary method for the shallow water equations on Cartesian grids: mass conservation and wetting and drying over variable topography
We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grids methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grids methods in weighted Wiener norms as well as analogues of the Littlewood-Paley-type characterizations in terms of families of quasi-interpolation operators.
Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space $VB_{p,q}^{δ,w}(\mathbb{R}^d)$ whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of $VB_{p,q}^{δ,w}(\mathbb{R}^d)$. Next, approximation formulas such as sparse grids are derived using the determined formula. The results of this study are expected to provide considerable insight into the …
