Will Pazner

Will Pazner, Per-Olof Persson

In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge-Kutta stages simultaneously. Numerical experiments are performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix-vector multiplications, and wall-clock time, for a wide range of time steps.

Will Pazner, Andrew Nonaka, John Bell et al.

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a multi-implicit spectral deferred correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework that allows us to simultaneously conserve mass and energy while satisfying on the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.

Will Pazner, Per-Olof Persson

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.

Will Pazner, Per-Olof Persson

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with that of traditional triangular elements. We solve the semi-discrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi's method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.

Will Pazner, Nathaniel Trask, Paul J. Atzberger

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows for compensating for differences in dissipative properties of discrete numerical operators relative to their continuum counter-parts. This allows us to handle general heterogeneous discretizations capturing accurately statistical relations. Our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.

Will Pazner

In this work, we construct high-order finite element spaces for the $L^2$ de Rham complex on triangular meshes amenable to low-order-refined preconditioning. The spaces are constructed using the Duffy transformation, by pulling back appropriately chosen polynomial spaces defined on the unit square; in addition to piecewise polynomials, these spaces also contain certain rational functions, and they reduce to the standard Lagrange, Nédélec, and discontinuous finite elements in the lowest-order case. We establish spectral equivalence, independent of the polynomial degree, of the stiffness matrices defined on these spaces with the lowest-order stiffness matrices defined on refined meshes, constructed using a Gauss-Lobatto triangular lattice. Spectral equivalence of the operators is a consequence of norm equivalences in Jacobi-weighted $L^2$ norms, which are established by proving stability of the Jacobi-Gauss-Lobatto interpolation operator in shifted norms. The low-order-refined preconditioners can also be used to precondition the standard piecewise polynomial finite element spaces using a fictitious space approach. The low-order-refined system can in turn be preconditioned effectively using algebraic multigrid methods. The analytical estimates are confirmed by numerical results on a variety of high-order problems, including on mixed meshes and surface meshes.

Stefanie Günther, Will Pazner, Dongping Qi

Based on the continuous interpretation of deep learning cast as an optimal control problem, this paper investigates the benefits of employing B-spline basis functions to parameterize neural network controls across the layers. Rather than equipping each layer of a discretized ODE-network with a set of trainable weights, we choose a fixed number of B-spline basis functions whose coefficients are the trainable parameters of the neural network. Decoupling the trainable parameters from the layers of the neural network enables us to investigate and adapt the accuracy of the network propagation separated from the optimization learning problem. We numerically show that the spline-based neural network increases robustness of the learning problem towards hyperparameters due to increased stability and accuracy of the network propagation. Further, training on B-spline coefficients rather than layer weights directly enables a reduction in the number of trainable parameters.

Will Pazner, Per-Olof Persson

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions.