arXiv subjects

Jason Hicken

Publications and source records attributed to Jason Hicken.

4 recordsLinked to original sources

arXiv2026

A collocation scheme that is equivalent to discontinuous Galerkin discretizations

A spectral collocation operator with the summation-by-parts property was introduced by Chan to develop entropy-stable discontinuous Galerkin (DG) semi-discretizations (https://doi.org/10.1016/j.jcp.2018.02.033). The present work shows that semi-discretizations based on this collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom in the collocation scheme are associated with the nullspace of the operator and remain zero throughout an unsteady simulation. If necessary, nullspace consistency can be recovered by introducing projection-based numerical dissipation that targets only the extraneous modes. The equivalence between collocation and DG solutions is verified for the constant-coefficient advection equation and Burgers' equation on triangular meshes. The numerical results show that equivalence breaks down for entropy-stable semi-discretizations of Burgers' equation based on a skew-symmetric splitting, but that equivalence can be recovered by projecting the collocation scheme's residual onto the relevant polynomial space. In addition to investigating equivalence, the results demonstrate that the collocation operator produces semi-discretizations with favorable spectral radii compared with a commonly used summation-by-parts operator construction.

math.NA↗

arXiv2024

Constructing stable, high-order finite-difference operators on point clouds over complex geometries

High-order difference operators with the summation-by-parts (SBP) property can be used to build stable discretizations of hyperbolic conservation laws; however, most high-order SBP operators require a conforming, high-order mesh for the domain of interest. To circumvent this requirement, we present an algorithm for building high-order, diagonal-norm, first-derivative SBP operators on point clouds over level-set geometries. The algorithm is not mesh-free, since it uses a Cartesian cut-cell mesh to define the sparsity pattern of the operators and to provide intermediate quadrature rules; however, the mesh is generated automatically and can be discarded once the SBP operators have been constructed. Using this temporary mesh, we construct local, cell-based SBP difference operators that are assembled into global SBP operators. We identify conditions for the existence of a positive-definite diagonal mass matrix, and we compute the diagonal norm by solving a sparse system of linear inequalities using an interior-point algorithm. We also describe an artificial dissipation operator that complements the first-derivative operators when solving hyperbolic problems, although the dissipation is not required for stability. The numerical results confirm the conditions under which a diagonal norm exists and study the distribution of the norm's entries. In addition, the results verify the accuracy and stability of the point-cloud SBP operators using the linear advection equation.

math.NA↗

arXiv2016

Conservative and Stable Degree Preserving SBP Operators for Non-Conforming Meshes

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new \emph{degree preserving} discretizations require an ansatz that the norm matrix of the SBP operator is of a degree $\geq 2p$, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is $2p-1$ accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.

math.NA↗

arXiv2015

Arnoldi-based Sampling for High-dimensional Optimization using Imperfect Data

We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The proposed sampling method is based on a generalization of Arnoldi's method used in Krylov iterative methods. We show that Arnoldi-based sampling can effectively estimate the dominant eigenvalues of the underlying Hessian, even in the presence of inaccurate gradients. This spectral information can be used to build a low-rank approximation of the Hessian in a quadratic model of the objective. We also investigate two variants of the linear term in the quadratic model: one based on step averaging and one based on directional derivatives. The resulting quadratic models are used in a trust-region optimization framework called the Stochastic Arnoldi's Method (SAM). Numerical experiments highlight the potential of SAM relative to conventional derivative-based and derivative-free methods when the design space is high-dimensional and noisy.

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