An adaptive multiresolution method on dyadic grids: Application to transport equations

We propose a modified adaptive multiresolution scheme for solving d$d$-dimensional hyperbolic conservation laws which is based on cell-average discretization in dyadic grids. Adaptivity is obtained by interrupting the refinement at the locations where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension. The tree structure allows us to succinctly represent the data and efficiently navigate through it. Dyadic grids also provide a more gradual refinement as compared with the traditional quad-trees (2D) or oct-trees (3D) that are commonly used for multiresolution analysis. We show some examples of adaptive binary tree representations, with significant savings in data storage when compared to quad-tree based schemes. As a test problem, we also consider this modified adaptive multiresolution method, using a dynamic binary tree data structure, applied to a transport equation in 2D domain, based on a second-order finite volume discretization.     

Wavelets and adaptive grids for the discontinuous Galerkin method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.     

A multigrid method with unstructured adaptive grids for steady Euler equations

The multigrid method based on multi-stage Jacobi relaxation, earlier developed by the authors for structured grid calculations with Euler equations, is extended to unstructured grid applications. The meshes are generated with Delaunay triangulation algorithms and are adapted to the flow solution.     

A study of high-order grid-characteristic methods on unstructured grids

In this paper, grid-characteristic methods for solving hyperbolic systems using approximation with high-order interpolation on unstructured tetrahedral and triangular grids are studied. Interpolation methods of order from 1 through 5 inclusive are considered. One-dimensional finite difference schemes for these methods are presented. The stability of these schemes is investigated. The grid-characteristic methods on unstructured triangular and tetrahedral grids are successfully used in seismic prospecting, specifically under the Arctic shelf and permafrost conditions. They are also used to solve problems of seismics, dynamic deformation and destruction, and to study anisotropic composite materials.     

The Galerkin method for singularly perturbed boundary value problems on adaptive grids

Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 5, pp. 138–148, September–October, 1990.     

A 3-D finite-volume method for the Navier-Stokes equations with adaptive hybrid grids

This paper describes the development and application of a numerical scheme for the three-dimensional Navier-Stokes equations of viscous flow using hybrid (prismatic/tetrahedral) grids. Employment of prisms is a relatively new approach towards complex geometry high Reynolds number viscous flow computations. The body surface is covered with triangles, which provides geometric flexibility, while the structure of the mesh in the direction normal to the surface provides better resolution of the viscous stresses. The irregular areas between different prismatic layers covering the surfaces of the domain are filled with tetrahedral elements. Their triangular faces match the corresponding triangular faces of the prisms. A dual adaptation scheme is developed which employs both directional and isotropic local refinement/coarsening of the prisms and tetrahedra. The structure of the prisms is preserved by avoiding interfaces with mid-edge nodes.

Spatial discretization consists of a finite-volume, node-based scheme that is of the central-differencing type. The solution is marched in time using a Taylor series expansion following the Lax-Wendroff approach. The scheme employs a dual cells arrangement for evaluation of the viscous terms, which has the property of suppressing odd-even modes in the solution. A storage-efficient data structure is employed, which utilizes the structure of the prismatic grid in one of the directions.     

Monotone high-order accurate compact scheme for quasilinear hyperbolic equations

Doklady Mathematics -     

Using Kronecker products to construct mimetic gradients

This paper describes how discrete versions of the derivative on the real line induce discrete version of the gradient and divergence in higher dimensions. This is geometrically motivated by results in algebraic graph theory since the grid in n dimensions is the graph Kronecker product of the path on n vertices. The resulting technique relies heavily on the matrix Kronecker product, and is an analogue of a derivation in multilinear algebra.     

Geometrically adaptive grids for stiff Cauchy problems

A new method for automatic step size selection in the numerical integration of the Cauchy problem for ordinary differential equations is proposed. The method makes use of geometric characteristics (curvature and slope) of an integral curve. For grids generated by this method, a mesh refinement procedure is developed that makes it possible to apply the Richardson method and to obtain a posteriori asymptotically precise estimate for the error of the resulting solution (no such estimates are available for traditional step size selection algorithms). Accordingly, the proposed methods are more robust and accurate than previously known algorithms. They are especially efficient when applied to highly stiff problems, which is illustrated by numerical examples.     

A direct method to construct triple systems

A triple system is a balanced incomplete block design D(v, k, λ, b, r) with k = 3. Although it has been shown that triple systems exist for all values of the parameters satisfying the necessary conditions:

$\text{λ(ν ? 1) ≡ 0 (}\text{mod 2}\text{), λν(ν ? 1) ≡ 0 (}\text{mod 6}\text{),}$

direct methods (nonrecursive) of construction are not available in general. In this paper we give a direct method to construct a triple system for all values of the parameters satisfying the necessary conditions.     

Spatially adaptive sparse grids for high-dimensional data-driven problems

Sparse grids allow one to employ grid-based discretization methods in data-driven problems. We present an extension of the classical sparse grid approach that allows us to tackle high-dimensional problems by spatially adaptive refinement, modified ansatz functions, and efficient regularization techniques. The competitiveness of this method is shown for typical benchmark problems with up to 166 dimensions for classification in data mining, pointing out properties of sparse grids in this context. To gain insight into the adaptive refinement and to examine the scope for further improvements, the approximation of non-smooth indicator functions with adaptive sparse grids has been studied as a model problem. As an example for an improved adaptive grid refinement, we present results for an edge-detection strategy.     

Wavelet-based adaptive grids for multirate partial differential-algebraic equations

The mathematical model of electric circuits yields systems of differential-algebraic equations (DAEs). In radio frequency applications, a multivariate model of oscillatory signals transforms the DAEs into a system of multirate partial differential-algebraic equations (MPDAEs). Considering quasiperiodic signals, an approach based on a method of characteristics yields efficient numerical schemes for the MPDAEs in time domain. If additionally digital signal structures occur, an adaptive grid is required to achieve the efficiency of the technique. We present a strategy applying a wavelet transformation to construct a mesh for resolving steep gradients in respective signals. Consequently, we employ finite difference methods to determine an approximative solution of characteristic systems in according grid points. Numerical simulations demonstrate the performance of the adaptive grid generation, where radio frequency signals with digital structures are resolved.     

Uniqueness of a high-order accurate bicompact scheme for quasilinear hyperbolic equations

The possibility of constructing new third- and fourth-order accurate differential-difference bicompact schemes is explored. The schemes are constructed for the one-dimensional quasilinear advection equation on a symmetric three-point spatial stencil. It is proved that this family of schemes consists of a single fourth-order accurate bicompact scheme. The result is extended to the case of an asymmetric three-point stencil.     

On families high-order accurate multioperator approximations of derivatives using two-point operators

A new family of multioperator approximations to derivatives of even and odd orders with inversion of two-point operators is considered. Existence and uniqueness theorems are stated for multioperators of formally arbitrary orders, and their spectral properties are examined. A scheme for a test hyperbolic equation with a multioperator approximation of 36th order is analyzed as an example. The accuracy and convergence of numerical solutions to the test problem are estimated.     

Using radial basis functions to construct local volatility surfaces

This article presents a new method for constructing a volatility surface for use in local volatility option pricing models. It builds on previous work focussing on non-parametric regression approaches using a set of radial basis functions, specifically thin plate splines. Optimal parameters are found using a trust region optimisation approach. While there is still much work to be done, the results are encouraging and show that the method is relatively tractable, stable and accurate.     

Analysis and computation of adaptive moving grids by deformation

We develop and analyze a numerical method for creating an adaptive moving grid in one-, two-, and three-dimensional regions. The method distributes grid nodes according to a given analytic or discrete weight function of the spatial and time variables, which reflects the fine structure of the solution. The weight function defines a vector field, which is used to construct a transformation of the computational domain into the physical domain. We prove that the resulting grid has the prescribed cell sizes and that no “mesh tangling” occurs. Numerical implementation of the method utilizes an efficient and robust least-squares solver to compute the vector field and a fourth-order Runge-Kutta scheme to determine the transformation. Results of several numerical experiments in one- and two-dimensions are also presented. These results indicate, among other things, that the method accurately redistributes the nodes and does not tangle the mesh. © 1996 John Wiley & Sons, Inc.     

Some notes on monotonicity preserving schemes on adaptive grids

By the example of the 1D transport equation with constant coefficient and the 2D linear transport equation with variable coefficients, the monotonization approach for two-step explicit finite-difference schemes is presented, based on studying scheme differential approximations [1,2]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)