Kiera van der Sande

Kiera van der Sande

Applied Math PhD Student at University of Colorado Boulder

Posts by Collection

portfolio

publications

Fast Variable Density 3-D Node Generation

Published in SIAM J. Sci. Comput., 2021

Mesh-free solvers for partial differential equations perform best on scattered quasi-uniform nodes. Computational efficiency can be improved by using nodes with greater spacing in regions of less activity. However, there is no ideal way to generate nodes for these solvers. We present an advancing front type method to generate variable density nodes in 2-D and 3-D with clear generalization to higher dimensions. The exhibited cost of generating a node set of size N in 2-D and 3-D with the present method is O(N)

Recommended citation: van der Sande, K. and Fornberg, B. (2021). “Fast Variable Density 3-D Node Generation).” SIAM J. Sci Comput.. 43(1). https://arxiv.org/pdf/1906.00636.pdf

Dynamic soliton-mean flow interaction with nonconvex flux

Published in Journal of Fluid Mechanics, 2021

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.

Recommended citation: van der Sande, K. and El, G. A. and Hoefer, A. (2021). “Dynamic soliton-mean flow interaction with non-convex flux.” J. Fluid Mech. 928(A21). https://www.cambridge.org/core/services/aop-cambridge-core/content/view/377B0DCF7BD95041734952601D5A4516/S002211202100803Xa_hi.pdf/dynamic-solitonmean-flow-interaction-with-non-convex-flux.pdf

Decreasing false-alarm rates in CNN-based solar flare prediction using SDO/HMI data

Published in The Astrophysical Journal Supplement Series, 2022

A hybrid two-stage machine-learning architecture that addresses the problem of excessive false positives (false alarms) in solar flare prediction systems is investigated. The first stage is a convolutional neural network (CNN) model based on the VGG-16 architecture that extracts features from a temporal stack of consecutive Solar Dynamics Observatory Helioseismic and Magnetic Imager magnetogram images to produce a flaring probability. The probability of flaring is added to a feature vector derived from the magnetograms to train an extremely randomized trees (ERT) model in the second stage to produce a binary deterministic prediction (flare/no-flare) in a 12 hr forecast window.

Recommended citation: Deshmukh, V. and Flyer, N. and van der Sande K. and Berger, T. (2022). “Decreasing false-alarm rates in CNN-based solar flare prediction using SDO/HMI data.” The Astrophysical Journal Supplement Series. 260(9). https://iopscience.iop.org/article/10.3847/1538-4365/ac5b0c/pdf

Fourier continuation discontinuous Galerkin methods for linear hyperbolic problems

Published in Communications on Applied Mathematics and Computation, 2022

Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. These methods have been used in PDE-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving stability and convergence. Here we propose the use of Fourier continuation in forming a new basis for the DG framework.

Recommended citation: van der Sande, K. and Appelo, D. and Albin, N. (2022). “Fourier continuation discontinuous Galerkin methods for linear hyperbolic problems”. Accepted 2022. https://arxiv.org/pdf/2105.00123.pdf

talks

teaching

Calculus I/II

Undergraduate courses for Engineers, University of Colorado Boulder, Department of Applied Mathematics, 2018

Differential Equations

Undergraduate courses for Engineers, University of Colorado Boulder, Department of Applied Mathematics, 2019