Jan S. Hesthaven

High-Order Collocation Methods for Differential Equations with Random Inputs

Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a high-order stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods.

Nodal discontinuous Galerkin methods on graphics processors

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. Lately, another property of DG has been growing in importance: The majority of a DG operator is applied in an element-local way, with weak penalty-based element-to-element coupling. The resulting locality in memory access is one of the factors that enables DG to run on off-the-shelf, massively parallel graphics processors (GPUs). In addition, DGs high-order nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG. Using a single US

Spectral methods for hyperbolic problems

400 Nvidia GTX 280 GPU, we accelerate a solver for Maxwells equations on a general 3D unstructured grid by a factor of around 50 relative to a serial computation on a current-generation CPU. In many cases, our algorithms exhibit full use of the devices available memory bandwidth. Example computations achieve and surpass 200gigaflops/s of net application-level floating point work. In this article, we describe and derive the techniques used to reach this level of performance. In addition, we present comprehensive data on the accuracy and runtime behavior of the method.

From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex

We review the current state of Fourier and Chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations. Throughout the discussion we emphasize recent developments in the area such as spectral penalty methods, the use of filters, the resolution of the Gibbs phenomenon, and issues related to the solution of nonlinear conservations laws such as conservation and convergence. We also include a brief discussion on the formulation of multi-domain methods for hyperbolic problems, and conclude with a few examples of the application of pseudospectral/collocation methods for solving nontrivial systems of conservation laws.

Spectral Methods for Time-Dependent Problems: Contents

The electrostatic interpretation of the Jacobi--Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi--Gauss--Lobatto quadrature points as the nodal sets are presented. The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss--Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h-p finite element methods.

Spectral Methods for Time-Dependent Problems: Index

Reference EPFL-BOOK-190435doi:10.1017/CBO9780511618352 URL: http://dx.doi.org/10.1017/CBO9780511618352 Record created on 2013-11-12, modified on 2017-05-12

A STABLE PENALTY METHOD FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS. I. OPEN BOUNDARY CONDITIONS

Reference EPFL-BOOK-190435doi:10.1017/CBO9780511618352 URL: http://dx.doi.org/10.1017/CBO9780511618352 Record created on 2013-11-12, modified on 2017-05-12

Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes

The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier--Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier--Stokes equations and utilizes linearization and localization at the boundaries based on these variables. The proposed boundary conditions are applied through a penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity. The versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder.

A Stable Penalty Method for the Compressible Navier--Stokes Equations: III. Multidimensional Domain Decomposition Schemes

Despite the popularity of high-order explicit Runge-Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems. The IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta method (ESDIRK). Furthermore, we apply adaptive time-step controllers based on the embedded temporal error predictors. We demonstrate in a number of numerical test problems that IMEX methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid-induced stiffness.

High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem

This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme. The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall. The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.