High-Order Adaptive and Parallel Discontinuous Galerkin Methods for Hyperbolic Conservation Laws J. E. Flaherty, L. Krivodonova, J. F. Remacle, and M. S. Shephard Scientific Computation Research Center

Discontinuous Galerkin Method Arbitrary order: extends finite volume method Structured or unstructured meshes No need for inter-element continuity – Simplifies adaptive h- and p-refinement

Discontinuous Galerkin Method Face-based communication – Simplifies parallel computation Sharp capturing of discontinuities Element level conservation A posteriori error estimates

Discontinuous Galerkin Method Face-based communication – Simplifies parallel computation Sharp capturing of discontinuities Element level conservation A posteriori error estimates However: More mesh unknowns than FEM for same order Possibly OK with parallel computation Monotonicty control (limiting) is difficult

DG Formulation Conservation law Construct a Galerkin problem on  j –cf. Cockburn and Shu (1989)

DG Solution Solving the Galerkin problem Integral evaluation Time integration Flux evaluation, limiting Approximation

Higher order equations Discontinuous approximations needs regularization for gradients

Example

Approximation u  U j  P j p  L 2 (  j ) –Orthogonal basis

Time Integration Explicit Runge-Kutta –TVB method of Cockburn and Shu (1989) –Local time stepping, Remacle et al. (2002) x t

Flux Evaluation Approximate f n (U j ) by a numerical flux F n (U j,U nb j ) –Define F n (U j,U nb j ) by a Riemann problem Possibilities: –Upwind: flux from inflow neighbor –Lax-Friedrichs: | max | is the maximum absolute eigenvalue of f u –Roe: linearized Riemann problem –van Leer: flux vector splitting –Colella-Woodward: contact surface resolution

Limiting Limiting: suppress spurious oscillations when p > 0 while maintaining order –Slope limiter: Cockburn and Shu (1989) –Curvature limiter: Barth (1990) –Moment limiter: Biswas et al. (1994) –Filtering: Gottlieb et al. (1999) No robust procedures for multi-dimensional situations

Slope vs. Moment Limiting Slope LimitingMoment Limiting Kinematic wave equation: u t + u x = 0 p = 2

Superconvergence One-dimensional conservation law Superconvergence at Radau points –Adjerid et al. (1995) –Biswas et al. (1994)

Superconvergence Theorem: If p > 0, the spatial discretization error of the DG method with U j  P p on [ x j-1,x j ] satisfies Proof: Use Galerkin orthogonality, properties of Legendre polynomials, and “strong” superconvergence at downwind element ends –cf. Adjerid et al. (2002)

A Posteriori Error Estimation One-dimensional conservation law DG method Error estimate

Superconvergence in p If f(u) = au: u t + au x = 0 –p = 0 –8 elements

Superconvergence in p If f(u) = au: u t + au x = 0 –p = 1 –8 elements

Superconvergence in p If f(u) = au: u t + au x = 0 –p = 2 –8 elements

Superconvergence in p If f(u) = au: u t + au x = 0 –p = 3 –8 elements

Superconvergence in p If f(u) = au: u t + au x = 0 –p = 4 –8 elements

Solitary Waves Nonlinear model: Exact solution: Solution at t = 1 Effectivity indices at t = 1

Two-Dimensional Problems Steady linear conservation law DG formulation –U  P p is the DG solution –   j - is the inflow boundary of  j –   j + outflow boundary of  j

Error Estimation Subtract the exact solution and use the Divergence Theorem Assume the error has a series expansion –h is a mesh parameter

Error Estimation Use an “induction” argument to show Orthogonal basis (on canonical triangle)

Error Estimation Show that –2D Radau polynomial? –Krivodonova and Flaherty (2001) Strong superconvergence

A Posteriori Error Estimation Solve the exit flow and DG problem –Complexity is O(p) per element

Example Consider –  = (0,1) x (0,1) –Exact solution

Error Estimates N p||e|| 0    04.85e e e e e e e e e e e e e e e

Superconvergence N pI-I- I+I+ I-I- I+I+ I-I- I+I e-11.07e-25.57e-25.08e-32.11e-22.47e e-37.53e-66.02e-48.95e-71.56e-41.09e e-51.98e-81.07e-54.8e e-61.8e e-68.3e e-76.2e e-85.4e-15

Linear Functional Barth and Larson (2002), Barth (2002) Functional

Superconvergence h 1/131/261/521/104 pI+I+ I+I+ rI+I+ rI+I+ r 01.72e-27.09e e e e-41.68e e e e-62.47e e e e-79.1e e-12118

Rayleigh-Taylor Instability Heavy fluid above a lighter one Subject to a perturbation DG method –Roe numerical flux –Limiting: reduce p by one if temperature becomes negative –Enrichment indicator 1/4 1/2

Adaptive Adaptive h-Refinement t= 1 t = 1.9t = 0.5t = 1.5

p-Refinement p = 0 p = 1 p = 2 p = 3 DOF

Vortex sheet

Blast

Perforated Shock Tube Mach 1.23 flow in tube Rupture a diaphragm between the tube and vent DG method with p = 0 Local time stepping Parallel computation Compare results with Nagamatsu et al. (1987)

Adaptive Analysis

Adaptivity

Discussion Stabilization (limiting) –Unstructured meshes Viscosity: Navier-Stokes equations Implicit time integration – Space-time DG method A posteriori error estimation –Discontinuities –Space-time coordination –Multi-dimensional superconvergence? Adaptive strategies –Optimal enrichment (with parallel computation)