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The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article. [ABSTRACT FROM AUTHOR]

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting. [ABSTRACT FROM AUTHOR]

The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article.

Compressed sensing (CS) algorithms mostly achieve linear array synthetic aperture radar (LASAR) 3D sparse imaging under the On-Grid strategy, they suffer from the mismatch measurement matrix and decreased imaging quality due to the position-errors among targets and preset scattering units. Aimed at this problem, based on the Off-Grid strategy, a sparse recovery algorithm via adaptive grids (SRAG) is proposed in this paper. First, the proposed algorithm extracts the target scattering units in the imaging scene under the On-Grid strategy. Second, the target scattering units with position-errors are extracted under the non-zero adjacent scattering coefficient criterion, and they are defined as the unfocused scattering units. Third, according to whether the unfocused scattering units' distance is smaller than the scattering unit spacing, their search-areas of targets' real positions are set, respectively, and differently. Finally, based on the minimum scattering coefficient residual criterion, we adaptively compensate the unfocused scattering units' position-errors and re-estimate their scattering coefficients. Both simulation and experimental results indicate that the SRAG algorithm decreases the position-errors effectively, it improves the imaging quality, estimation accuracy of targets' positions, and scattering coefficients than traditional On-Grid CS algorithms. [ABSTRACT FROM AUTHOR]

In this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Equ 28:955, 2016). For small spatial grid-sizes, we establish some useful Fredholm properties for the operator that arises after linearizing our system around the travelling wave solutions to the original Nagumo PDE. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is a spatially stretched and twisted version of the standard second order differential operator that is associated to the PDE waves. [ABSTRACT FROM AUTHOR]

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations. [ABSTRACT FROM AUTHOR]

The paper presents two methods to improve the efficiency of supersonic flow simulation using arbitrarily shaped unstructured grids. The first method promotes increasing the numerical solution convergence rate and is based on the geometric multigrid method for initialization of the flow field. The method is used to obtain the initial field of distributed physical quantity values, which maximally corresponds to the converged solution. For this purpose, the problem simulation is performed on a series of coarse grids beginning from the coarsest one in this series. Upon completion of simulations, the solution obtained is interpolated to a finer grid and used for initialization of simulations on this grid. The second method allows increasing the numerical solution accuracy and is based on statically adapting the computational grid to the flow specifics. The static adaptation algorithm provides automatic refinement of the computational grid in the region of specific features of flow, such as shock waves typical for supersonic flows. This algorithm provides a better description of the shock-wave front owing to the local grid refinement, with the local refinement region being automatically selected. Results of using these methods are demonstrated for the two supersonic aerodynamics problems: the simulation of the bow shock strength at a given distance under axially symmetric body Seeb-ALR and a mock-up aircraft Lockheed Martin 1021. It is shown that in both cases, the numerical solution convergence rate is increased owing to the use of the geometric multigrid method for initialization and a higher quality and a higher accuracy of solution is gained owing to the local grid refinement (using static adaptation means) near the shock-wave front. [ABSTRACT FROM AUTHOR]

In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case. [ABSTRACT FROM AUTHOR]

Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision of the adaptive computations, while reducing CPU time and memory requirements. Implementation issues of the open source code CARMEN-MHD are discussed. To illustrate its precision and efficiency different benchmark computations including shock-cloud interaction and magnetic reconnection are presented. [ABSTRACT FROM AUTHOR]

We introduce a novel liquid simulation approach that combines a spatially adaptive pressure projection solver with the Particle-in-Cell (PIC) method. The solver relies on a generalized version of the Finite Difference (FD) method to approximate the pressure field and its gradients in tree-based grid discretizations, possibly non-graded. In our approach, FD stencils are computed by using meshfree interpolations provided by a variant of Radial Basis Function (RBF), known as RBF-Finite-Difference (RBF-FD). This meshfree version of the FD produces differentiation weights on scattered nodes with high-order accuracy. Our method adapts a quadtree/octree dynamically in a narrow-band around the liquid interface, providing an adaptive particle sampling for the PIC advection step. Furthermore, RBF affords an accurate scheme for velocity transfer between the grid and particles, keeping the system's stability and avoiding numerical dissipation. We also present a data structure that connects the spatial subdivision of a quadtree/octree with the topology of its corresponding dual-graph. Our data structure makes the setup of stencils straightforward, allowing its updating without the need to rebuild it from scratch at each time-step. We show the effectiveness and accuracy of our solver by simulating incompressible inviscid fluids and comparing results with regular PIC-based solvers available in the literature. [ABSTRACT FROM AUTHOR]

A technique is presented for numerical studying of the liquid jet impact on the surface layer of an elastic-plastic body under conditions close to cavitation damage. The technique comprises three stages. First, assuming that the body is perfectly rigid, the shock-wave dynamics in the jet and surrounding gas is calculated and the law of pressure variation on the body surface is derived. Then, the derived law is approximated by splines and simple analytic expressions. The obtained approximation is used to determine the fields of stresses in the surface layer of the body and deformations of its surface. The proposed approach is illustrated by computing the impact of high-speed water jet on the rigid body out of nickel alloy.

This study concerns the asymptotic stability of an eikonal, or ray, transformation based Peaceman–Rachford splitting method for solving the paraxial Helmholtz equation with high wave numbers. Arbitrary nonuniform grids are considered in transverse and beam propagation directions. The differential equation targeted has been used for modeling propagations of high intensity laser pulses over a long distance without diffractions. Self-focusing of high intensity beams may be balanced with the de-focusing effect of created ionized plasma channel in the situation, and applications of grid adaptations are frequently essential. It is shown rigorously that the fully discretized oscillation-free decomposition method on arbitrary adaptive grids is asymptotically stable with a stability index one. Simulation experiments are carried out to illustrate our concern and conclusions. [ABSTRACT FROM AUTHOR]

This paper deals with the traditional cartesian mesh to make them body-fitted based on computer graphics and local mesh refinement according to the variation of flow. Combining with the characteristics of rectangular grids and unstructured grid, the adaptive body-fitted Cartesian mesh generation program is written and the cut-cell DSMC method is developed based on it. In order to improve the computational efficiency, some methods like local simulated molecules and dynamic time steps are used too. Numerical tests indicate that the proposed method greatly improves the search efficiency in the progress of calculation, at the same time, it also ensures body-fitted grids, high calculation accuracy and easy implementation for adaptive grids. [ABSTRACT FROM AUTHOR]

Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic partial differential equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries. The grid refinement is flexible and adaptive. The discretization is combined with a fast solution method, which incorporates asynchronous time stepping adapted to the spatial scale. The framework is validated on linear problems in curved and unbounded domains. Key applications include the obstacle problem and the one-phase Stefan free boundary problem. [ABSTRACT FROM AUTHOR]

In this paper, a modified central finite difference scheme for a three-point nonuniform grid is presented for the one-dimensional homogeneous Helmholtz equation using the Bloch wave property. The modified scheme provides highly accurate solutions at the nodes of the nonuniform grid for very small to very large range of wave numbers irrespective of how the grid is adapted throughout the domain. A variety of numerical examples are considered to validate the superiority of the modified scheme for a nonuniform grid over a standard central finite difference scheme. [ABSTRACT FROM AUTHOR]

Modeling and visualization of a rainwater overland flow is an important tool for a risk assessment, preparation, evacuation planning, and real-time forecasting of flood warning. The objective of this research is to develop a numerical software to visualize the rainwater overland flow based on a finite volume method for shallow water equations and in combination with the dynamically adaptive tree grid technique and the dynamic domain-defining method. The obtained simulations for several experiments were tested and compared with results in literature, both theoretical and experimental results. The comparisons with non-adaptive grids show that the developed algorithm for simulation is very efficient and has a potential for practical usages, in terms of computational times and accuracy. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

When performing shallow flow simulations on adaptive grids, the C-property (i.e. conservation property) and the mass conservation may not be simultaneously preserved, that is, either C-property or mass conservation is likely to be violated following grid refining or coarsening. The cause of such a contradiction is analyzed in detail in this work, which essentially links to the reconstruction of bed and flow information in those newly created cells during grid adaptation. An effective approach is subsequently proposed to resolve the contradiction by locally modifying the bed elevation in certain problematic cells when reconstructing flow information using linear interpolation as part of the grid adaptation procedure. [ABSTRACT FROM AUTHOR]

This paper explores numerical errors that arise when large-eddy simulation (LES) is used with adaptive mesh refinement (AMR). LES and AMR combined can reduce the computational cost of turbulence simulations compared to direct numerical simulations, but are rarely used together due to complications that arise with the application of the turbulence closure model at different grid resolutions. Errors appear at grid refinement interfaces due to dependence of computed quantities on the LES filter width and insufficient smoothness of the solution at the grid scale. Here, explicit filtering and approximate reconstruction of the unfiltered velocity field are used to mitigate the effects of these errors in a simulation of decaying isotropic turbulence advected past a grid refinement interface. In particular, different explicit filter types and levels of reconstruction are tested. Explicit filtering with zero-level reconstruction is found to produce the best long-term convergence to a uniform grid solution with minimum perturbation at the interface. Higher levels of reconstruction yield better near-interface convergence. When explicit filtering is used, the explicit filter width transition is more important than the grid spacing transition in terms of solution convergence and interface perturbation. These results inform the use of LES on more complicated AMR grids. [ABSTRACT FROM PUBLISHER]

An adaptive grid method which redistributes grid points according to equidistribution principle was implemented to an in-house computational fluid dynamics code capable of simulating blast waves. The resultant code was first tested for a shock tube problem. It was observed that benefit of using adaptive grids becomes more evident when discontinuities in the flow are stronger. It was also observed that interpolation method used to move the flow variables to new grid locations directly affects the accuracy of the solution and interpolation methods which do not guarantee conservation of mass may yield highly inaccurate results. Blast wave simulations performed showed that the adaptive grid method used here improved predictions considerably without requiring a lot of extra CPU time. [ABSTRACT FROM AUTHOR]

Abstract: This paper illustrates the use of an adaptive finite element method to solve a non-linear singularly perturbed boundary value problem which arises from a one-dimensional -tensor model of liquid crystals. The adaptive non-uniform mesh is generated by equidistribution of a strictly positive monitor function which is a linear combination of a constant floor and a power of the first derivative of the numerical solution. By an appropriate selection of the monitor function parameters, we show that the computed numerical solution converges at an optimal rate with respect to the mesh density and that the solution accuracy is robust to the size of the singular perturbation parameter. [Copyright &y& Elsevier]

Abstract: In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes. [Copyright &y& Elsevier]

Abstract: This paper describes some applications to three-dimensional problems of a comprehensive grid generation method that relies on inverted Beltrami equations in control metrics. The method enables one to generate in a unified manner both fixed and adaptive grids in domains and on surfaces with complicated geometry and/or multi-scaled physical quantities. The applications are related to the solutions of singularly perturbed equations with interior and boundary layers and to the solution of the heat equation in a two-phased medium modelling heat transfer between a solid material and an embedded thermocouple. [Copyright &y& Elsevier]

nalytical approaches to investigating the asymptotic disposition for a special iterative discrete-stochastic algorithm for constructing adaptive grids using the Kohonen self-organizing maps are analyzed. A 'recurrent' approach to obtaining the most probable mean positions of grid nodes for a small number of iterations is developed for a simplified one-dimensional (1D) case. This approach allows interesting analytical investigations and numerical testing for the algorithm considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with the efficient solution of the linear systems of equations that arise from an adaptive space-implicit time discretisation of the Black-Scholes equation. These nonsymmetric systems are very large and sparse, so an iterative method will usually be the method of choice. However, such a method may require a large number of iterations to converge, particularly when the timestep used is large (which is often the case towards the end of a simulation which uses adaptive timestepping). An appropriate preconditioner is therefore desirable. In this paper we show that a very simple multigrid algorithm with standard components works well as a preconditioner for these problems. We analyse the eigenvalue spectrum of the multigrid iteration matrix for uniform grid problems and illustrate the method's efficiency in practice by considering the results of numerical experiments on both uniform grids and those which use adaptivity in space. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes the algorithms and functionality of a new module developed to support overset grid assembly associated with performing time-dependent and adaptive moving body calculations of external aerodynamic flows using a multi-solver paradigm (i.e. different CFD solvers in different parts of the computational domain). We use the term “domain connectivity” in this paper to denote all the procedures that are involved in an overset grid assembly, and the module developed is referred henceforth as the domain-connectivity module. The domain-connectivity module coordinates the data transfer between different solvers applied in different parts of the computational domain – body fitted structured or unstructured to capture viscous near-wall effects, and Cartesian adaptive mesh refinement to capture effects away from the wall. The execution of the CFD solvers and the domain-connectivity module are orchestrated by a Python-based computational infrastructure. The domain-connectivity module is fully parallel and performs all its operations (identification of grid overlaps and determination of data interpolation strategy) on the partitioned grid data. In addition, the domain connectivity procedures are completely automated such that no user intervention or manual input is necessary. The capabilities and performance of the package are presented for several test problems, including flow over a NACA 0015 wing and an AGARD A2 slotted airfoil, hover simulation of a scaled V-22 rotor, and dynamic simulation of a UH-60A rotor in forward flight. A modification to the algorithm for improved domain connectivity solutions in problems with tight tolerances as well as heterogeneous grid clustering is also presented. [Copyright &y& Elsevier]

In this manuscript, we develop a multilevel framework for the pricing of a European call option based on multiresolution techniques. In this approach, the Black–Scholes equation is transformed via finite differences into a system of linear equations, where the form of the implicit operator is used to construct coarse grid projectors. The reduction of the computational resource is achieved by truncating small wavelet coefficients. However, because traditional wavelets fail to prevent oscillations from developing in the Greeks, a multilevel approach is used to retain smoothness in Gamma by incorporating derivative information into the multiresolution analysis. [ABSTRACT FROM AUTHOR]

Fast solvers for Poisson's equation with boundary conditions at infinity are an important building block for molecular dynamics. One issue that arises when this equation is solved numerically is the infinite size of the domain. This prevents a direct solution so that other concepts have to be considered. Within this paper a method is discussed that employs hierarchically coarsened grids to overcome this problem. Special attention has to be paid to the discretization at the grid interfaces. A finite volume approach is used for the same. The resulting set of linear equations is solved using a fast-adaptive composite grid algorithm. Emphasis is put on the implementation of the method on the STI cell broadband engine, a modern multi core processor, that is powerful in floating point operations and memory bandwidth. Code optimization techniques are applied as well as parallelization of the code to get maximum performance on this processor. For validation of the performance test runs are executed and the runtime is analyzed in detail. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Construction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The extended Numerov scheme of Chawla is one of high-order compact finite-difference discretisations applicable to BVPs in second-order ODEs, and to PDEs. The use of such discretisations in connection with adaptive grids remains largely unexplored. In order to make up this deficiency, we study the combination of the extended Numerov discretisation with the iterative local adaptive grid h-refinement previously tested by the present author at the assumption of the conventional finite-difference discretisation. A detailed formalism enabling a posteriori error analysis of the extended Numerov scheme is developed, together with several alternative choices for error estimators, grid refinement indicators, and regridding strategies. The adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed ODEs, including equations relevant for electrochemistry. The most satisfactory algorithm involves: deferred approach to error estimation, based on virtual grid coarsening for truncation error evaluation; local discretisation errors as refinement indicators; and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement. The extended Numerov scheme proves more efficient than the conventional discretisation, when used together with the iterative h-refinement. [Copyright &y& Elsevier]

Abstract: Despite many years of the development of adaptive grid methods, there is still a need for re-evaluation, tuning, and comparisons of the various approaches to grid adaptation. In this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement/de-refinement algorithm for the finite-difference solution of singularly perturbed BVPs in second-order ODEs, that is also easily extendable to evolutionary PDEs. Conventional three-point discretisations of the first and second derivatives are used. Errors of the discrete solution and its first derivative are simultaneously controlled. The grid refinement is accomplished by adding nodes at inter-nodal positions, and de-refinement by removing previously added nodes in reverse order. Computational experiments are performed for 15 examples of ODEs, assuming several alternative choices of the a posteriori error estimators, grid refinement indicators, and regridding strategies. The most satisfactory results are obtained by combining: deferred approach to error estimation, based on higher-order reference derivative approximations for truncation error evaluation; interpolation-based indicator for grid refinement; and regridding strategy that uses the mean indicator value as the indicator threshold for refinement. [Copyright &y& Elsevier]

This article illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples. [ABSTRACT FROM AUTHOR]

Recently, we reported a novel method for integrating the time dependent Schrödinger equation which used hydrodynamic quantum trajectories to adapt the boundaries of a fixed spatial grid. The moving boundary truncation (MBT) method significantly reduced the number of grid points needed to perform accurate calculations while maintaining stability during the time propagation. In this work, the method is extended to multidimensional examples. The application of MBT to scattering on 2D and 3D potential energy surfaces shows a greater decrease in the number of grid points needed compared with full fixed grids while maintaining excellent accuracy and stability, even for very long propagation times. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [ABSTRACT FROM AUTHOR]

In the early days of computing, geophysical fluid dynamics (GFD), predominantly numerical weather prediction (NWP), was a dominant factor in the design of computer architecture and algorithms. This early work focused initially on finite difference algorithms on rectangular computational grids and later on spectral methods. After the initial work of von Neumann, Charney, and Arakawa however, the focus shifted from the basic algorithms to improvements in the physical models. Further work on fundamental numerical algorithms shifted to other disciplines—predominately the then emerging aerospace community. As a result, for 40 years, the GFD community has been using numerical techniques that are virtually unchanged. The two primary numerical methodologies that have been used for modeling the atmosphere and the ocean have been spectral methods for global modeling and structured rectilinear grids for regional modeling. In 1992, work began on the Operational Multiscale Environment model with Grid Adaptivity (OMEGA), a new atmospheric simulation tool based upon an adaptive unstructured grid. Over the past 14 years, this model has found application to atmospheric dispersion, point weather forecasting, and, the topic of this paper, hurricane track forecasting. The unstructured grid paradigm employed in OMEGA has the advantage of flexibility in providing high resolution where required by either static physical properties (terrain elevation, coastlines, land use) or the changing dynamical situation. This paper provides a description of this new paradigm and presents its use in atmospheric simulation. [ABSTRACT FROM AUTHOR]

Abstract: Electrochemical kinetic simulations by the patch-adaptive strategy outlined in the previous parts of this series of papers can be computationally expensive in the extreme cases of very thin boundary layers, moving fronts, or models defined over multiple spatial subintervals having disparate scales. The replacement of the second-order accurate conventional spatial discretisation (thus far used by the strategy) by the fourth-order accurate extended Numerov discretisation of Chawla [M.M. Chawla, J. Inst. Math. Appl. 22 (1978) 89], discussed in this work, allows one to reduce the computational costs. Numerical experiments with five representative examples of difficult-to-simulate kinetic models reveal that the reduction of the computational time, the number of spatial grid nodes needed, and spatial grid refinement levels, is particularly noticeable when a high spatial accuracy of the simulations is requested. At low accuracy demands the conventional spatial discretisation may be more efficient and more convenient to use. The patch-adaptive simulations by the extended Numerov scheme require spatial tolerance parameter TOLX values that are several times larger than the TOLX values that ensure a comparable accuracy by the conventional spatial discretisation. [Copyright &y& Elsevier]

A new regularization method, adaptive grid regularization, has been presented. Numerical results there show in a convincing way that this method is a powerful tool to identify discontinuities of solutions of ill-posed problems. It is the aim of this paper to give a convergence analysis for this new method. [ABSTRACT FROM AUTHOR]

In this paper we apply the new regularization method a daptive grid regularization, which is well-suited for ill-posed problems with discontinuous solutions and which was introduced in an earlier paper [5], to 2D linear integral equations of the first kind. After describing the method we present some numerical examples showing that this method is a powerful tool to identify discontinuities in ill-posed problems. [ABSTRACT FROM AUTHOR]

Abstract: The cost-effectiveness of moving mesh adaptation is studied in a number of 1D tests. We propose a method that is based on two established modern techniques. First, we use a moving mesh approach based on the classic equidistribution method. Second, we discretize the model equations for grid and physics using a conservative finite volume method and we solve the resulting equations with a preconditioned inexact Newton–Krylov method. Using these state of the art methods, we consider the question of whether a real improvement in performance can be achieved using adaptive grids. We consider rigorous metrics of the accuracy and cost of a numerical solution on uniform and adaptive grids. For a number of classic but challenging problems we demonstrate that indeed adaptive grids can lead to a great improvement in cost-effectiveness. [Copyright &y& Elsevier]

Results of calculations of fluid flow over a step located on a channel bottom are given. Numerical modeling is performed for the model of free-boundary potential flows of an ideal incompressible fluid using a finite-difference method with dynamically adaptive grids. The behavior of the free surface in the neighborhood of the step is studied as a function of the incident-flow velocity. The results are compared with experimental data. [ABSTRACT FROM AUTHOR]

Confident adaptive algorithms are described, evaluated, and compared with other algorithms that implement the estimation of motion. A Galerkin finite element adaptive approach is described for computing optical flow, which uses an adaptive triangular mesh in which the resolution increases where motion is found to occur. The mesh facilitates a reduction in computational effort by enabling processing to focus on particular objects of interest in a scene. Compared with other state-of-the-art methods in the literature our adaptive methods show only motion where main movement is known to occur, indicating a methodological improvement. The mesh refinement, based on detected motion, gives an alternative to methods reported in the literature, where the adaptation is usually based on a gradient intensity measure. A confidence is calculated for the detected motion and if this measure passes the threshold then the motion is used in the adaptive mesh refinement process. The idea of using the reliability hypothesis test is straightforward. The incorporation of the confidence serves the purpose of increasing the optical flow determination reliability. Generally, the confident flow seems most consistent, accurate and efficient, and focuses on the main moving objects within the image. © 2006 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 16, 35–50, 2006 [ABSTRACT FROM AUTHOR]

In this paper, a computational method for solving fluid flow problems with moving interfaces is presented. Herein, adaptive quadtree grids are used coupled with the CICSAM [O. Ubbink, Numerical prediction of two fluid systems with sharp interfaces, PhD Thesis, Imperial College of Science, Technology and Medicine, London, 1997] free surface capturing volume of fluid (VoF) method and PLIC reconstruction to interpolate the volume fraction field during refinement and derefinement processes. The combination of high resolution adaptive hierarchical remeshing and CICSAM interface advection is shown to overcome the problems of interface smearing and high CPU intensivity inherent in most VoF schemes. The result is a combination of free surface tracking and free surface capturing in that the interface is effectively tracked by the adapting refinements in the quadtree grid. In this way, a sharp interface is achieved and the advantages of both free surface tracking and capturing are combined. The new method is applied to interface advection examples in translating, rotating and shearing flow fields, and the benefits of using adapting quadtree grids demonstrated. [Copyright &y& Elsevier]

This paper considers a new adaptive wavelet solver for two-dimensional systems based on an adaptive block refinement (ABR) method that takes advantage of the quadtree structure of dyadic blocks in rectangular regions of the plane. The computational domain is formed by non-overlapping blocks. Each block is a uniform grid, but the step size may change from one block to another. The blocks are not predetermined, but they are dynamically constructed according to the refinement needs of the numerical solution. The decision over whether a block should be refined or unrefined is taken by looking at the magnitude of wavelet coefficients of the numerical solution on such block. The wavelet coefficients are defined as differences between values interpolated from a coarser level and known function values at the finer level. The main objective of this paper is to establish a general framework for the construction and operation on such adaptive block-grids in 2D. The algorithms and data structure are formulated by using abstract concepts borrowed from quaternary trees. This procedure helps in the understanding of the method and simplifies its computational implementation. The ability of the method is demonstrated by solving some typical test problems. [Copyright &y& Elsevier]

After a brief review of various moving grid methods for resolution enhancement, we present the deformation method originated from differential geometry. The method is used to relocate the nodes of an initial grid according to the solution through a set of deformation differential equations. The node velocity is determined by a monitor function constructed from physical variable(s) through a scalar Poisson equation. It is proved mathematically that the cell volume is proportional to the monitor function at each time step. In particular, the moving grid mapping has positive Jacobian determinant and thus will not fold into itself. The moving grid method is then applied to the Euler equation. The results showed that the grid indeed follows the monitor function closely and that the method significantly improves the resolution compared to uniform grids of the same amount of nodes. [ABSTRACT FROM AUTHOR]