Finite-volume and adaptive numerical methods for nonlinear hyperbolic PDEs, motivated by transport-dominated systems in geophysical and environmental modeling.
Overview
In my research I focus on the development of robust numerical methods for nonlinear hyperbolic partial differential equations, with an emphasis on transport-dominated systems arising in geophysical and environmental applications.
My work in this area combines numerical analysis, algorithm design, and large-scale computation, often motivated by challenges encountered in applied geophysical and environmental modeling. This work provides the mathematical foundation for several applied and interdisciplinary research areas described elsewhere on this site.
Above: Coordinate system for the two-layer shallow water system demonstrating internal waves, from (Mandli, 2013).
Right: Riemann solver demonstrating water overtopping a zero-width barrier on sloping bathymetry, from (Li & Mandli, 2021).
Core Themes
Finite-volume methods for conservation laws
Treatment of dry states, wetting/drying, and sharp interfaces
Adaptive mesh refinement for multiscale solutions
Stability and accuracy in complex geometries
Transport-dominated and nonsmooth solution behavior
Methods & Approaches
Key methodological contributions include:
Finite-volume schemes for shallow water and multilayer flow models
Adaptive mesh refinement strategies for hyperbolic systems
Cut-cell and barrier methods for complex domains
Transport-aware approaches to reduced-order modeling
This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.
2021
An \h\-Box Method for Shallow Water Equations Including Barriers
When approximating a function that depends on a parameter, one encounters many practical examples where linear interpolation or linear approximation with respect to the parameters proves ineffective. This is particularly true for responses from hyperbolic partial differential equations (PDEs) where linear, low-dimensional bases are difficult to construct. We propose the use of displacement interpolation, where the interpolation is done on the optimal transport map between the functions at nearby parameters, to achieve an effective dimensionality reduction of hyperbolic phenomena. We further propose a multidimensional extension by using the intertwining property of the Radon transform. This extension is a generalization of the classical translational representation of Lax and Phillips [P. D. Lax and R. S. Phillips, Bull. Amer. Math. Soc., 70 (1964), pp. 130–142].
2016
Clawpack: building an open source ecosystem for solving hyperbolic PDEs
Kyle T Mandli, Aron J Ahmadia, Marsha Berger, Donna Calhoun, David L George, and 4 more authors
Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.
2013
A numerical method for the two layer shallow water equations with dry states
A numerical method is proposed for solving the two layer shallow water equations with variable bathymetry in one dimension based on high-resolution f-wave-propagation finite volume methods. The method splits the jump in the fluxes and source terms into waves propagating away from each grid cell interface. It addresses the required determination of the system’s eigenstructure and a scheme for evaluating the flux and source terms. It also handles dry states in the system where the bottom layer depth becomes zero, utilizing existing methods for the single layer solution and handling single layer dry states that can exist independently. Sample results are shown illustrating the method and its handling of dry states including an idealized ocean setting.
2012
PyClaw: Accessible, Extensible, Scalable Tools for Wave Propagation Problems
David I. Ketcheson, Kyle Mandli, Aron J. Ahmadia, Amal Alghamdi, Manuel Quezada de Luna, and 3 more authors
Development of scientific software involves tradeoffs between ease of use, generality, and performance. We describe the design of a general hyperbolic PDE solver that can be operated with the convenience of MATLAB yet achieves efficiency near that of hand-coded Fortran and scales to the largest supercomputers. This is achieved by using Python for most of the code while employing automatically wrapped Fortran kernels for computationally intensive routines, and using Python bindings to interface with a parallel computing library and other numerical packages. The software described here is PyClaw, a Python-based structured grid solver for general systems of hyperbolic PDEs [K. T. Mandli et al., PyClaw Software, Version 1.0, http://numerics.kaust.edu.sa/pyclaw/ (2011)]. PyClaw provides a powerful and intuitive interface to the algorithms of the existing Fortran codes Clawpack and SharpClaw, simplifying code development and use while providing massive parallelism and scalable solvers via the PETSc library. The package is further augmented by use of PyWENO for generation of efficient high-order weighted essentially nonoscillatory reconstruction code. The simplicity, capability, and performance of this approach are demonstrated through application to example problems in shallow water flow, compressible flow, and elasticity.
