Workshop on Scientific Computing

Abstracts: It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature. In this talk, we report on our recent development of a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in the literature, with the main ingredients being summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy stable flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Numerical experiments will be reported to validate the accuracy and shock capturing efficacy of these entropy stable DG methods. This is a joint work with Tianheng Chen.

Abstracts: TBA

Abstracts: In this talk we shall discuss the local discontinuous Galerkin methods with the generalized upwind numerical flux and generalized alternating numerical fluxes, when solving one- and/or two-dimensional convection diffusion problems. Firstly the periodic condition is considered and the global L²-norm error estimate is optimal. To do that, we will introduce the generalized Gauss-Radau (GGR) projection, which is defined globally, not locally. Secondly the singularity perturbation of convection diffusion equation with Dirichlet condition is considered, which has a stationary outflow boundary layer. The local L²-norm error estimate is double-optimal, namely, not only the width of pollution domain is optimal, and also the order of L²-norm error out of pollution domain is optimal. To this purpose, we will introduce suitable weight functions nearby the outflow boundary, and carefully establish a deep investigation on the GGR projection.

Abstracts: In this talk we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modelled by stochastic differential equations (SDEs). We propose a new numerical integrator based on a stochastic splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem can be viewed as a perturbation. We provide rigorous error analysis for the new numerical integrator using the backward error analysis technique and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow problems of physical interests.

Abstracts: Ideal magnetohydrodynamic (MHD) equations are widely used in many areas in physics and engineering, and these equations have a divergence-free constraint on the magnetic field. In this paper, we propose high order globally divergence-free numerical methods to solve the ideal MHD equations. The algorithms are based on discontinuous Galerkin methods in space. The magnetic induction equations are discretized separately to approximate the normal components of the magnetic field on elements interfaces, and to extract additional information about the magnetic field when higher order accuracy is desired. This is then followed by an element by element reconstruction to obtain the globally divergence-free magnetic field. In time, strong-stability-preserving Runge-Kutta methods are applied. In consideration of accuracy and stability of the methods, a careful investigation is carried out, both numerically and analytically, to study the choices of the numerical fluxes associated with the electric field at element interfaces and vertices. The resulting methods are local and the approximated magnetic fields are globally divergence-free. Numerical examples are presented to demonstrate the accuracy and robustness of the methods.

Abstracts: For a parabolic problem with maximal L^p-regularity, we prove that the time discretization by a linear multistep method or Runge-Kutta method preserves the maximal L^p-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge-Kutta methods of all orders preserve maximal regularity. In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the L^q norm, 1 <= q <= \infty, and the maximal L^p-regularity of finite element solutions of parabolic equations are proved.

Abstracts: A brief introduction of translation-invariant kernel based methods, aka radial basis function methods, for function approximation and PDEs will be given. These methods do not require meshes, but in return, we must deal with highly ill-conditioned linear systems. In this talk, we will introduce an adaptive algorithm that selects appropriate column subspaces that ensure linear independency.

Abstracts: In this talk, we present semi-implicit spectral deferred correction (SDC) methods for hyperbolic systems of conservation laws with stiff relaxation terms. The relaxation term is treated implicitly, and the convection terms are treated by explicit schemes. The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit and can be constructed easily and systematically for any order of accuracy. Weighted essentially non-oscillatory (WENO) schemes are adopted in spatial discretization to achieve high accuracy in space. After a description of the mathematical properties of the schemes, several applications will be presented to demonstrate the capability of the schemes. This is a joint work with C. Sun.

Abstracts: In this talk, we consider waveform relaxation (WR) methods for solving time-dependent PDEs. WR methods are distinctive in that a typical subdomain problem is posed in both space and time; each iteration requires the parallel solution of these space-time subproblems, followed by an exchange of interface data defined over the whole time window. An often cited advantage of WR methods is that they allow each subdomain to use a different spatial and temporal grid that is adapted to the dynamics of the local subproblem.

Here, we show that WR methods have another advantage, namely that they allow additional parallelism in time. The key observation is that in most cases, one has enough initial and interface data to start an WR iteration even before the previous iteration has completed. In other words, several iterations can run simultaneously. Based on this observation, we propose two ways of parallelizing WR methods in time. The first one uses a fixed time window and is mathematically equivalent to the original WR method. The second one chooses the time-window size dynamically based on how many free processors are available; this leads to a method with different convergence behaviour. We demonstrate the effectiveness of both approaches by comparing their running times against those obtained from classical time-stepping methods, where the same number of processors is used to parallelize in space only.

Abstracts: We propose a framework for proving unconditional stability and convergence of Crank-Nicolson ADI schemes for two-dimensional Riesz space-fractional diffusion equations with non-separable variable coefficients in the sense of discrete L2 norm, in which the Crank-Nicolson discretization and an unspecified Toeplitz discretization are employed to approximate the first-order temporal derivative and the fractional order spatial derivative, respectively. This is the first attempt to propose Crank-Nicolson ADI schemes for the equation with non-separable variable coefficients. To the best of our knowledge, unconditional stability (convergence) of previous ADI schemes for the equation are established only for single-spatial-variate-dependent-separable coefficients (spatially-independent coefficients). In contrast, we show the unconditional stability and convergence of Crank-Nicolson ADI schemes for the equation with non-separable three-variates functions (one temporal variate and two spatial variates) under certain mild assumptions on the diffusion coefficients and the unspecified Toeplitz discretization. The assumptions imposed on the unspecified Toeplitz discretization are satisfied by all existing schemes. The assumptions imposed on the diffusion coefficients are related to partial Lipschitz continuity, which is weaker than Lipschitz continuity. Moreover, a splitting preconditioner is employed to precondition the one-dimensional systems resulting from the propose ADI schemes, with which the condition numbers of resulting preconditioned matrices are uniformly bounded by a constant independent of discretization step-sizes. As a result, the ADI systems are solved withing an optimal arithmetic complexity. Numerical results are reported to support the theoretical analysis and to show the efficiency of the proposed method.