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Item A Pseudo-Differential Sweeping Method for the Helmholtz Equation(2024-04-02) Johnson, Raven Shane; Chan, JesseShow more Ultrasound-guided medical procedures often experience complications when imaging heterogeneous tissue. Computer simulations of the ultrasound field offer a workable solution to this heterogeneity problem, but the computational methods required for these simulations tend to be either highly accurate and computationally slow or computationally quick and inaccurate. We propose a sweeping numerical method for solving the Helmholtz equation which is built from a truncated pseudo-differential expansion. We discretize this expansion using high order spectral element methods in space and an explicit time-stepping method in time. Numerical experiments examine the behavior of the proposed method in 1D and 2D under different numerical parameters. We demonstrate that the proposed sweeping method is not only accurate but increases in accuracy as the angular frequency increases.Show more Item A short note on a Bernstein-Bezier basis for the pyramid(Society for Industrial and Applied Mathematics, 2016) Chan, Jesse; Warburton, T.Show more We introduce a Bernstein--Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein--Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties common to Bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature. Procedures for differentiation and integration of these basis functions are also discussed.Show more Item Bernstein-Bézier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media(2018-10-10) Guo, Kaihang; Chan, JesseShow more Efficient and accurate simulations of wave propagation are central to applications in seismology. In practice, heterogeneities arise from the presence of different types of rock in the subsurface. Additionally, simulations over long time periods require high order approximation to avoid numerical dispersion and dissipation effects. The weight-adjusted discontinuous Galerkin (WADG) method delivers high order accuracy for arbitrary heterogeneous media. However, the cost of WADG grows rapidly with the order of approximation. To reduce the computational complexity of high order methods, we propose a Bernstein-Bézier WADG method, which takes advantage of the sparse structure of matrices under the Bernstein-Bézier basis. Our method reduces the computational complexity from O(N^6) to O(N^4) in three dimensions and is highly parallelizable to implement on Graphics Processing Units (GPUs).Show more Item A Comparison of High Order Interpolation Nodes for the Pyramid(Society for Industrial and Applied Mathematics, 2015) Chan, Jesse; Warburton, T.Show more The use of pyramid elements is crucial to the construction of efficient hex-dominant meshes [M. Bergot, G. Cohen, and M. Duruflé, J. Sci. Comput., 42 (2010), pp. 345--381]. For conforming nodal finite element methods with mixed element types, it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra. We adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes, including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron [T. Warburton, J. Engrg. Math., 56 (2006), pp. 247--262]. Comparisons between nodal sets show that the lowest Lebesgue constants are given by warp and blend nodes for order $N\leq 7$ and Fekete nodes for $N>7$, though numerical experiments show little variation in the conditioning and accuracy of all surveyed nonequidistant points.Show more Item Efficient computation of Jacobian matrices for entropy stable summation by parts schemes(2021-06-22) Taylor, Christina Gabrielle; Chan, JesseShow more This work presents efficient formulas for the computation of Jacobian matrices arising from entropy stable summation-by-parts schemes. Competing methods for computing Jacobian matrices include finite difference, automatic differentiation, graph coloring, and Jacobian-free Newton-Krylov methods. In contrast to these methods the formulas proposed provide a sparsity-informed method for computing Jacobian matrices that are free of truncation error. Computational timings confirm that the proposed formulas scale very robustly with respect to the size of the system and easily outperform existing methods on denser Jacobian matrices. Two applications of Jacobian matrices, two-derivative and implicit time stepping, are also explored in numerical experiments with Burgers' and the shallow water equations.Show more Item Energy and Entropy Stable High-Order Discontinuous Galerkin Methods on Cut Meshes(2024-08-09) Taylor, Christina; Chan, JesseShow more In this work we present two contributions to cut mesh methods: a provably energy stable discontinuous Galerkin (DG) method with state redistribution and an entropy stable DG scheme for hyperbolic conservation laws. Cut meshes schemes use a simple unfitted background mesh of standard elements that is then cut by embedded boundaries to yield a hybrid mesh of cut and uncut elements. While these methods are a convenient means of mesh generation for complex geometries, the arbitrary size and shape of cut elements pose two significant challenges: the small cell problem--where the arbitrary size/shape of cut elements can severely restrict the CFL condition--and the construction of stable and accurate schemes on cut elements. We address the small cell problem using state redistribution and prove it can be combined with an energy stable scheme without damaging energy stability. For entropy stability, we construct hybridized summation-by-parts operators to reduce guaranteeing entropy stability to constructing exact, positive weight quadrature rules on cut elements. We construct such quadrature on cut elements using subtriangulations and Carathéodory pruning. We numerically verify the high-order accuracy and energy/entropy conservation and stability of our schemes using the acoustic wave, shallow water, and compressible Euler equations.Show more Item Entropy Stable Discontinuous Galerkin-Fourier Methods(2020-09-23) Lin, Yimin; Chan, JesseShow more Entropy stable discontinuous Galerkin methods for nonlinear conservation laws replicate an entropy inequality at semi-discrete level. The construction of such methods depends on summation-by-parts (SBP) operators and flux differencing using entropy conservative finite volume fluxes. In this work, we propose a discontinuous Galerkin-Fourier method for systems of nonlinear conservation laws, which is suitable for simulating flows with spanwise homogeneous geometries. The resulting method is semi-discretely entropy conservative or entropy stable. Computational efficiency is achieved by GPU acceleration using a two-kernel splitting. Numerical experiments in 3D confirm the stability and accuracy of the proposed method.Show more Item High order entropy stable discontinuous Galerkin methods for the shallow water equations: networks, quasi-1D flows, and positivity preservation(2022-05-09) Wu, Xinhui; Chan, JesseShow more High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. They have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This thesis focuses on high order entropy stable discontinuous Galerkin schemes for the shallow water equations, and extends them in two primary directions. First, we extend high order entropy stable discontinuous Galerkin method for nonlinear conservation laws to both multi-dimensional domains and networks constructed from 1D domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment. Next, we provide both continuous and semi-discrete entropy analyses for the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. Each quasi-1D formulation includes additional terms to account for varying channel or nozzle widths. These extra terms correspond to non-conservative terms in the original equations, and introduce asymmetry into the numerical fluxes. We design new entropy conservative fluxes for both sets of equations and supply proofs of entropy conservation on periodic quasi-1D domains. Furthermore, we show that the new entropy conservative fluxes for the shallow water equations are well-balanced for continuous bathymetry profiles. Finally, we present a high order entropy stable discontinuous Galerkin method for nonlinear shallow water equations on 2D triangular meshes which preserves the positivity of the water height for constant bathymetry. The scheme combines a low order invariant domain preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well-balanced for meshes which are fitted to continuous bathymetry profiles. Finally, we apply the proposed method to a realistic large scale simulation of the 1959 Malpasset Dam break to verify the robustness of the scheme.Show more Item High order weight-adjusted discontinuous Galerkin methods for problems in wave propagation(2021-04-30) Guo, Kaihang; Chan, JesseShow more Efficient and accurate simulations of wave propagation have a wide range of applications in science and engineering, from seismic and medical imaging to rupture and earthquake simulations. In this thesis, we focus on provably energy stable, efficient and high order accurate discontinuous Galerkin (DG) methods for wave problems with general choices of basis and quadrature. We first extend existing DG solvers for the acoustic and elastic wave equations to coupled elastic-acoustic media. A simple upwind-like numerical flux is derived to weakly impose continuity of the normal velocity and traction at elastic-acoustic interfaces. When paired with the weight-adjusted DG (WADG) method, the resulting scheme is consistent and energy stable on curvilinear meshes and for arbitrary heterogeneous media, including anisotropy and sub- cell heterogeneities. We also present an application of this coupled DG solver to an inverse problem in photoacoustic tomography (PAT). Then, we develop DG methods for wave equations on moving meshes. In the proposed method, an arbitrary Lagrangian-Eulerian (ALE) formulation is adopted to map the acoustic wave equation from the time-dependent moving physical domain to a fixed reference domain. The spatially varying geometric terms produced by the ALE transformation are approximated using a weight-adjusted DG discretization. The resulting semi-discrete WADG scheme is provably energy stable up to a term which converges to zero with the same rate as the optimal L2 error estimate. Finally, two mesh regularization approaches are proposed to address the possible mesh tangling caused by boundary deformations. We use the Bernstein triangular representation to convert a curved mesh into the Bernstein control net which only consists of linear sub-triangles. The validity of curved meshes is guaranteed when all sub-triangles are untangled. Our first approach borrows the idea of the spring-mass system for adaptation of first-order linear meshes. The second approach solves an optimization problem to prevent the appearance of small Jacobian determinants of each sub-triangle in the control net.Show more Item High Order, Entropy Stable, Positivity Preserving Discontinuous Galerkin Discretizations of Compressible Flow(2023-11-29) Lin, Yimin; Chan, JesseShow more High order DG methods offer improved accuracy in turbulent and under-resolved flows due to low numerical dissipation and dispersion, but may lose robustness due to the lack of stabilization. A priori stabilization techniques, such as artificial viscosity, slope limiting, and filtering, require heuristic parameter, which does not provide provable guarantees of robustness and may lead to over-dissipation of solutions. For the compressible Euler and Navier-Stokes equations, physically meaningful solutions must maintain positive physical quantities and satisfy entropy stability. The primary objective of this dissertation is to develop high order DG methods that are provably entropy stable and preserve positivity through a posteriori limiting techniques We first introduce a more general discretization of the viscous terms in the compressible Navier-Stokes equations, which also enables a simple and explicit imposition of entropy stable wall boundary conditions. A positivity limiting strategy for entropy-stable discontinuous Galerkin spectral element (ESDGSEM) discretizations is then introduced. The strategy is constructed by blending high order solutions with a low order positivity-preserving and semi-discretely entropy stable discretization through an elementwise limiter. The proposed limiting strategy is both semi-discretely entropy stable and positivity preserving for the compressible Navier-Stokes equations under an appropriate CFL condition. Another approach we propose is an entropy stable limiting strategy for discontinuous Galerkin spectral element (DGSEM) discretizations. The strategy is an extension to the subcell limiting strategy that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We discuss the extension of the proposed subcell limiting strategy to preserve positivity and general convex constraints. Numerical experiments confirm the high order accuracy, entropy stability, and robustness of the proposed strategies.Show more Item Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion(Elsevier, 2018) Chan, Jesse; Evans, John A.Show more We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces results in less stringent CFL restrictions than equivalentﾠC0ﾠor discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based onﾠL2n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods.Show more Item Numerical methods for boundary integral equations(2020-08-13) Zhang, Yabin; Gillman, Adrianna; Chan, Jesse; Riviere, Beatrice; Stanciulescu, IlincaShow more The thesis focuses on numerical methods for boundary integral equation (BIE) formulations of partial differential equations (PDEs). The work contains three parts: the first two consider numerical solution methods for boundary integral equations in wave scattering and Stokes flow, respectively. The last part proposes an adaptive discretization technique for BIEs in 2D. The proposed work is based on previous developments in fast direct solution techniques for BIEs. Such methods exploit the rank deficiency in the off-diagonal blocks of the discretized system and build an approximation to the inverse with linear cost for two-dimensional problems. Once the inverse approximation is constructed, applying it to any given vector is very cheap, making the methods ideal for problems with lots of right-hand-sides. The two direct solvers presented in this thesis are driven by real-life applications. The scattering solver is built to assist practitioners in designing acoustic and optic devices to manipulate waves. Its efficiency in handling multiple incident angles and minor modifications in the structure will be handy in an optimal design setting. The Stokes solver is to help with numerical simulation of objects such as bacteria and vesicles in viscous flow. To accurately capture the interaction between the objects and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the objects. This makes standard fast direct solvers too expensive to be useful, as the linear system changes for each time step. The proposed approach avoids this by pre-constructing a fast direct solver for the wall independently of time and updating the original solver to accommodate any refinements in discretization. The last part of the thesis describes an adaptive discretization technique for two-dimensional BIEs. Standard adaptive discretization method often requires a sequence of global boundary density solves each on a finer grid and terminates with the last grid if the improvements obtained from the next finer level is very small. The global density solves make the cost of the standard approach relatively high. The proposed alternative reduces the cost by replacing global solves with local solves for an approximate of the true density.Show more Item On discretely entropy conservative and entropy stable discontinuous Galerkin methods(Elsevier, 2018) Chan, JesseShow more High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEsﾠ[1],ﾠ[2]. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matricesﾠ[3],ﾠ[4],ﾠ[5],ﾠ[6]. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.Show more Item On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows(Frontiers Media S.A., 2022) Chan, Jesse; Ranocha, Hendrik; Rueda-Ramírez, Andrés M.; Gassner, Gregor; Warburton, TimShow more High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.Show more Item Reduced storage nodal discontinuous Galerkin methods on semi-structured prismatic meshes(Elsevier, 2017) Chan, Jesse; Wang, Zheng; Hewett, Russell J.; Warburton, T.Show more We present a high order time-domain nodal discontinuous Galerkin method for wave problems on hybrid meshes consisting of both wedge and tetrahedral elements. We allow for vertically mapped wedges which can be deformed along the extruded coordinate, and present a simple method for producing quasi-uniform wedge meshes for layered domains. We show that standard mass lumping techniques result in a loss of energy stability on meshes of vertically mapped wedges, and propose an alternative which is both energy stable and efficient. High order convergence is demonstrated, and comparisons are made with existing low-storage methods on wedges. Finally, the computational performance of the method on Graphics Processing Units is evaluated.Show more Item Simulations of Partially Miscible Two-Component Two-Phase Flow at the Pore-Scale Using Discontinuous Galerkin Methods(2020-10-02) Lin, Lu; Riviere, Beatrice; Heinkenschloss, Matthias; Chapman, Walter; Chan, JesseShow more In this dissertation, an effective numerical algorithm is developed for establishing simulation for the two-component two-phase flow with partial miscibility at the pore scale. Many studies in the rock-fluid interaction have been done for immiscible flow, whose components do not mix and separate instantaneously. This paper extends the study to miscible flow, whose components will mix with certain pressure and temperature, and exploits the potential of simulating complex real-life fluid interactions. The mathematical model consists of a set of Cahn-Hilliard equations and a realistic equation of state (i.e. Peng-Robinson equation of state). The numerical challenges lie in the fact that these are highly coupled, fourth-order, nonlinear partial differential equations. For solving the proposed PDEs, a discontinuous Galerkin (DG) method is used for space discretization, and a combination of backward Euler method and convex-concave splitting method is used for time discretizition. The resulting simulation can extract essential characteristics of the digital rock sample, agreeing with conventional lab-based tests but with only a fraction of cost in time and resources. Practically, the proposed algorithm and simulation can help engineers to make more informed decisions, for example in oil industry for enhancing oil recovery.Show more Item Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media(Wiley, 2018) Chan, JesseShow more Weight‐adjusted inner products are easily invertible approximations to weighted L2 inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.Show more