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Cavitation is a process where bubbles form and collapse within a fluid with dynamic, spatially varying pressure. This phenomenon can concentrate energy density by 12 orders of magnitude, creating light-emitting plasma or damaging nearby surfaces. A key question in cavitation theory and experiments is: what are the upper limits of energy density achievable through this spontaneous multiscale process? Among the many physical processes at play, we focus on fluid compressibility, modeled using the Tait-Murnaghan equation of state for a homentropic Euler fluid. We examine spherical cavities corresponding to experimentally realizable sonoluminescing bubbles, whose radius changes by a factor of over 100. These bubbles reach velocities exceeding the speed of sound of the surrounding fluid. However, all-Mach hydrodynamic solvers, such as those implemented using the Basilisk software, can exhibit unphysical behavior even at early times when motion is nearly incompressible. To accurately capture high Mach number motion and resolve dynamics in the sonoluminescence regime, we introduced a uniform bubble approximation for the ideal gas inside the bubble. This leading-order approximation clarifies the significant effects of compressibility. Our results reproduce the equation-of-state-dependent asymptotic power-law region predicted by analytic calculations. This confirms our method's ability to capture high Mach number motion and suggests that the asymptotic regime could be experimentally observed. Convergence of this method is demonstrated for bubbles in both water and liquid lithium, showing that compressibility slows collapse. Additionally, an outgoing shock wave in the compressible fluid is resolved., Comment: 19 pages, 9 figures

We present and study techniques for investigating the spectra of linear differential operators on surfaces and flat domains using symmetric meshfree methods: meshfree methods that arise from finding norm-minimizing Hermite-Birkhoff interpolants in a Hilbert space. Meshfree methods are desirable for surface problems due to the increased difficulties associated with mesh creation and refinement on curved surfaces. While meshfree methods have been used for solving a wide range of partial differential equations (PDEs) in recent years, the spectra of operators discretized using radial basis functions (RBFs) often suffer from the presence of non-physical eigenvalues (spurious modes). This makes many RBF methods unhelpful for eigenvalue problems. We provide rigorously justified processes for finding eigenvalues based on results concerning the norm of the solution in its native space; specifically, only PDEs with solutions in the native space produce numerical solutions with bounded norms as the fill distance approaches zero. For certain problems, we prove that eigenvalue and eigenfunction estimates converge at a high-order rate. The technique we present is general enough to work for a wide variety of problems, including Steklov problems, where the eigenvalue parameter is in the boundary condition. Numerical experiments for a mix of standard and Steklov eigenproblems on surfaces with and without boundary, as well as flat domains, are presented, including a Steklov-Helmholtz problem.

We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show a general result relating boundedness, solvability, and convergence that can be used to find eigenvalues. The method works by extending a solution to a surface PDE into a box-shaped domain so that the differential operators of the extended function agree with the surface differential operators, as in the Closest Point Method. This differs from approaches that require a basis for the surface of interest, which may not be available. Numerical experiments are also provided, demonstrating super-algebraic convergence. Current high-order methods for surface PDEs are often limited to a small class of surfaces or use radial basis functions (RBFs). Our approach offers certain advantages related to conditioning, generality, and ease of implementation. The method is meshfree and works on arbitrary surfaces (closed or non-closed) defined by point clouds with minimal conditions.

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold's interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM's finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver. We demonstrate our method's convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations., Comment: 26 pages

We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface {construction methods} yield comparable patterns to those {observed} in real-life animals., Comment: 22 pages, 16 figures

The DD-CPM software library provides a set of tools for the discretization and solution of problems arising from the closest point method (CPM) for partial differential equations on surfaces. The solvers are built on top of the well-known PETSc framework, and are supplemented by custom domain decomposition (DD) preconditioners specific to the CPM. These solvers are fully compatible with distributed memory parallelism through MPI. This library is particularly well suited to the solution of elliptic and parabolic equations, including many reaction-diffusion equations. The software is detailed herein, and a number of sample problems and benchmarks are demonstrated. Finally, the parallel scalability is measured., Comment: Accepted for publication in Numerical Algorithms

The discretization of surface intrinsic PDEs has challenges that one might not face in the flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the flat space to their closest points on the surface. This mapping brings intrinsic data onto the embedding space, allowing us to numerically approximate PDEs by the standard methods in the tubular neighborhood of the surface. Here, we solve the surface intrinsic positive Helmholtz equation by the CPM paired with finite differences which usually yields a large, sparse, and non-symmetric system. Domain decomposition methods, especially Schwarz methods, are robust algorithms to solve these linear systems. While there have been substantial works on Schwarz methods, Schwarz methods for solving surface differential equations have not been widely analyzed. In this work, we investigate the convergence of the CPM coupled with Schwarz method on 1-manifolds in d-dimensional space of real numbers., Comment: To appear in the proceedings of the 26th Domain Decomposition meeting in Hong Kong

In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this article, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods., Comment: 32 pages, 17 figures

In \cite{ZH2019}, we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favourably to the inverse Lax-Wendroff method in \cite{TS2010,TWSN2012} where analysis and numerical experiments have previously verified the presence of order reduction \cite{TS2010,TWSN2012}. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all non-negative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws., Comment: To appear in Journal of Computational Physics. arXiv admin note: text overlap with arXiv:1908.01027

The discretization of surface intrinsic elliptic partial differential equations (PDEs) poses interesting challenges not seen in flat space. The discretization of these PDEs typically proceeds by either parametrizing the surface, triangulating the surface, or embedding the surface in a higher dimensional flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the embedding space to their closest points on the surface. In the CPM, this mapping also serves as an extension operator that brings surface intrinsic data onto the embedding space, allowing PDEs to be numerically approximated by standard methods in a narrow tubular neighborhood of the surface. We focus here on numerically approximating the positive Helmholtz equation, $\left(c-\Delta_\mathcal{S}\right)u=f,~c\in\mathbb{R}^+$ by the CPM paired with finite differences. This yields a large, sparse, and non-symmetric system to solve. Herein, we develop restricted additive Schwarz (RAS) and optimized restricted additive Schwarz (ORAS) solvers and preconditioners for this discrete system. In particular, we develop a general strategy for computing overlapping partitions of the computational domain, as well as defining the corresponding Dirichlet and Robin transmission conditions. We demonstrate that the convergence of the ORAS solvers and preconditioners can be improved by using a modified transmission condition where more than two overlapping subdomains meet. Numerical experiments are provided for a variety of analytical and triangulated surfaces. We find that ORAS solvers and preconditioners outperform their RAS counterparts, and that using domain decomposition as a preconditioner gives faster convergence over using it as a solver, as expected. The methods exhibit good parallel scalability over the range of process counts tested., Comment: Accepted to SISC

The discretization of elliptic PDEs leads to large coupled systems of equations. Domain decomposition methods (DDMs) are one approach to the solution of these systems, and can split the problem in a way that allows for parallel computing. Herein, we extend two DDMs to elliptic PDEs posed intrinsic to surfaces as discretized by the Closest Point Method (CPM) \cite{SJR:CPM,CBM:ICPM}. We consider the positive Helmholtz equation $\left(c-\Delta_\mathcal{S}\right)u = f$, where $c\in\mathbb{R}^+$ is a constant and $\Delta_\mathcal{S}$ is the Laplace-Beltrami operator associated with the surface $\mathcal{S}\subset\mathbb{R}^d$. The evolution of diffusion equations by implicit time-stepping schemes and Laplace-Beltrami eigenvalue problems \cite{CBM:Eig} both give rise to equations of this form. The creation of efficient, parallel, solvers for this equation would ease the investigation of reaction-diffusion equations on surfaces \cite{CBM:RDonPC}, and speed up shape classification \cite{Reuter:ShapeDNA}, to name a couple applications., Comment: To appear in the proceedings of the 25th Domain Decomposition meeting in Saint John's Newfoundland

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF- FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231(14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.

Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds ${\mathcal M}$ and ${\mathcal N}$ to the simpler problems of solving a PDE on ${\mathcal M}$ and projecting to the closest points on ${\mathcal N}.$ In our approach, an embedding PDE is formulated in the embedding space using closest point representations of ${\mathcal M}$ and ${\mathcal N}.$ This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes $p$-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and $p$-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance colour images., Comment: 19 pages, 6 figures

We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory.

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities., Comment: 25 pages, 9 figures

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.

Sonoluminescence is the phenomena of light emission from a collapsing gas bubble in a liquid. Theoretical explanations of this extreme energy focusing are controversial and difficult to validate experimentally. We propose to use molecular dynamics simulations of the collapsing gas bubble to clarify the energy focusing mechanism, and provide insight into the mechanism of light emission. In this paper, we model the interior of a collapsing noble gas bubble as a hard sphere gas driven by a spherical piston boundary moving according to the Rayleigh-Plesset equation. We also include a simple treatment of ionization effects in the gas at high temperatures. By using fast, tree-based algorithms, we can exactly follow the dynamics of million particle systems during the collapse. Our results clearly show strong energy focusing within the bubble, including the formation of shocks, strong ionization, and temperatures in the range of 50,000---500,000 degrees Kelvin. Our calculations show that the gas-liquid boundary interaction has a strong effect on the internal gas dynamics. We also estimate the duration of the light pulse from our model, which predicts that it scales linearly with the ambient bubble radius. As the number of particles in a physical sonoluminescing bubble is within the foreseeable capability of molecular dynamics simulations we also propose that fine scale sonoluminescence experiments can be viewed as excellent test problems for advancing the art of molecular dynamics.