Margarete Oliveira Domingues

An adaptive multiresolution scheme with local time stepping for evolutionary PDEs

We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on a second-order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure. Numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time-stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrate the additional speed-up of the local time stepping for a thermo-diffusive flame instability.

Advantage of wavelet technique to highlight the observed geomagnetic perturbations linked to the Chilean tsunami (2010)

The vertical component (Z) of the geomagnetic field observed by ground-based observatories of the International Real-Time Magnetic Observatory Network has been used to analyze the induced magnetic fields produced by the movement of a tsunami, electrically conducting sea water through the geomagnetic field. We focus on the survey of minutely sampled geomagnetic variations induced by the tsunami of 27 February 2010 at Easter Island (IPM) and Papeete (PPT) observatories. In order to detect the tsunami disturbances in the geomagnetic data, we used wavelet techniques. We have observed an 85% correlation between the Z component variation and the tide gauge measurements in period range of 10 to 30 min which may be due to two physical mechanisms: gravity waves and the electric currents in the sea. As an auxiliary tool to verify the disturbed magnetic fields, we used the maximum variance analysis (MVA). At PPT, the analyses show local magnetic variations associated with the tsunami arriving in advance of sea surface fluctuations by about 2 h. The first interpretation of the results suggests that wavelet techniques and MVA can be effectively used to characterize the tsunami contributions to the geomagnetic field and further used to calibrate tsunami models and implemented to real-time analysis for forecast tsunami scenarios.

Study of local regularities in solar wind data and ground magnetograms

Abstract Interplanetary coronal mass ejections (ICMEs) can reach the Earth׳s magnetosphere causing magnetic disturbances. For monitoring purposes, some satellites measure the interplanetary parameters which are related to energy transfer from solar wind into magnetosphere, while ground-based magnetometers measure the geomagnetic disturbance effects. Data from the ACE satellite and from some representative magnetometers were examined here via discrete wavelet transform (DWT). The increase in the amplitude of wavelet coefficients of solar wind parameters and geomagnetic field data is well-correlated with the arrival of the shock and sheath regions, and the sudden storm commencement and main phase, respectively. As an auxiliary tool to verify the disturbed magnetic fields identified by the DWT, we developed a new approach called effectiveness wavelet coefficient (EWC) methodology. The first interpretation of the results suggests that DWT and EWC can be effectively used to characterize the fluctuations on the solar wind parameters and their contributions to the geomagnetic field. Further, this kind of technique could be implemented in quasi real-time to facilitate the identification of the shock and the passage of the sheath region which sometimes can be followed by geoeffective magnetic clouds. Also, the technique shows to be very useful for the identification of time intervals in the dataset during geomagnetic storms which are associated to interplanetary parameters under very well defined conditions. It allows selecting ideal events for investigation of magnetic reconnection in order to highlight in a more precise manner the mechanisms existing in the electrodynamical coupling between the solar wind and the magnetosphere.

Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

In this paper, we discuss the use of the sparse point representation (SPR) methodology for adaptive finite-difference simulations in computational electromagnetics. The principle of the SPR method is to represent the solution only through those point values indicated by the significant wavelet coefficients, which are used as local regularity indicators. Recently, two kinds of SPR schemes have been considered for solving Maxwells equations: 1) staggered grids in the time-space domain are used for the discretization of the magnetic and electrical fields, as in the finite-differences time-domain (FDTD) scheme and 2) nonstaggered grids are used in combination with Runge-Kutta ODE solvers. In both cases, 1-D simulations of the SPR method leads to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time with substantial gain in memory and computational speed. However, in the latter case, we found spurious oscillations in the simulations. Therefore, before extending the implementation of the SPR method to higher dimensions, we wanted to evaluate which of these two SPR strategies is more convenient. After a careful theoretical analysis of stability and numerical dispersion comparing the schemes in staggered and nonstaggered grids, we conclude that schemes for staggered grids seem to be preferable from the dispersion viewpoint, especially for low-order schemes and coarse grids. However, by adapting the grid density and increasing the order, SPR schemes for nonstaggered grids also show good performance. In our experiments, no spurious oscillations were detected. We observed that, for a given accuracy, the adaptive scheme on a nonstaggered grid requires less computational effort. Since the use of nonstaggered grids increases the stability range and facilitates the implementation of adaptive strategies, we believe that the SPR method in nonstaggered grids has a very good potential for computational electromagnetics

Adaptive wavelet representation and differentiation on block-structured grids

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.

Daubechies wavelet coefficients: a tool to study interplanetary magnetic field fluctuations

We have studied a set of 41 magnetic clouds (MCs) measured by the ACE spacecraft, using the discrete orthogonal wavelet transform (Daubechies wavelet of order two) in three regions: Pre-MC (plasma sheath), MC and Post-MC. We have used data from the IMF GSM-components with time resolution of 16 s. The mathematical property chosen was the statistical mean of the wavelet coefficients (⟨Dd1⟩). The Daubechies wavelet coefficients have been used because they represent the local regularity present in the signal being studied. The results reproduced the well- known fact that the dynamics of the sheath region is more than that of the MC region. This technique could be useful to help a specialist to find events boundaries when working with IMF datasets, i.e., a best form to visualize the data. The wavelet coefficients have the advantage of helping to find some shocks that are not easy to see in the IMF data by simple visual inspection. We can learn that fluctuations are not low in all MCs, in some cases waves can penetrate from the sheath to the MC. This methodology has not yet been tested to identify some specific fluctuation patterns at IMF for any other geoeffective interplanetary events, such as Co-rotating Interaction Regions (CIRs), Heliospheric Current Sheet (HCS) or ICMEs without MC signatures. In our opinion, as is the first time that this technique is applied to the IMF data with this purpose, the presentation of this approach for the Space Physics Community is one of the contributions of this work.

Comparison of Adaptive Multiresolution and Adaptive Mesh Refinement Applied to Simulations of the Compressible Euler Equations

We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in space with second order shock-capturing, and explicit time integration either with or without local time-stepping. The two methods are benchmarked for the compressible Euler equations in Cartesian geometry. As test cases a 2D Riemann problem, Lax-Liu 6, and a 3D ellipsoidally expanding shock wave have been chosen. We compare and assess their computational efficiency in terms of CPU time and memory requirements. We evaluate the accuracy by comparing the results of the adaptive computations with those obtained with the corresponding FV scheme using a regular fine mesh. We find that both approaches yield similar trends for CPU time compression for increasing number of refinement levels. MR exhibits more efficient memory compression than AMR and shows slightly enhanced convergence; however, a larger absolute overhead is measured for the tested codes.

The discrete complex wavelet approach to phase assignment and a new test bed for related methods

A new approach based on the dual-tree complex wavelet transform is introduced for phase assignment to non-linear oscillators, namely, the Discrete Complex Wavelet Approach-DCWA. This methodology is able to measure phase difference with enough accuracy to track fine variations, even in the presence of Gaussian observational noise and when only a single scalar measure of the oscillator is available. So, it can be an especially interesting tool to deal with experimental data. In order to compare it with other phase detection techniques, a testbed is introduced. This testbed provides time series from dynamics similar to non-linear oscillators, such that a theoretical phase choice is known in advance. Moreover, it allows to tune different types of phase synchronization to test phase detection methods under a variety of scenarios. Through numerical benchmarks, we report that the proposed approach is a reliable alternative and that it is particularly effective compared with other methodologies in the presence of moderate to large noises.

Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwells equation numerical solutions.

An adaptive multiresolution method for ideal magnetohydrodynamics using divergence cleaning with parabolic-hyperbolic correction

We present an adaptive multiresolution method for the numerical simulation of ideal magnetohydrodynamics in two space dimensions. The discretization uses a finite volume scheme based on a Cartesian mesh and an explicit compact Runge-Kutta scheme for time integration. Hartens cell average multiresolution allows to introduce a locally refined spatial mesh while controlling the error. The incompressibility of the magnetic field is controlled by using a Generalized Lagrangian Multiplier (GLM) approach with a mixed hyperbolic-parabolic correction. Different applications to two-dimensional problems illustrate the properties of the method. For each application CPU time and memory savings are reported and numerical aspects of the method are discussed. The accuracy of the adaptive computations is assessed by comparison with reference solutions computed on a regular fine mesh. Adaptive multiresolution methods for solving the MHD equations using divergence cleaning yield excellent results for the test cases presented.Hartens thresholding strategy gives the most efficient results.The MR computations reduce significantly the CPU time and the memory requirements with respect to a regular grid, while maintaining the precision.