Murtazo Nazarov

A SECOND-ORDER MAXIMUM PRINCIPLE PRESERVING LAGRANGE FINITE ELEMENT TECHNIQUE FOR NONLINEAR SCALAR CONSERVATION EQUATIONS ∗

This paper proposes an explicit, (at least) second-order, maximum principle sat- isfying, Lagrange finite element method for solving nonlinear scalar conservation equations. The technique is based on a new viscous bilinear form introduced in Guermond and Nazarov (Com- put. Methods Appl. Mech. Engrg., 272 (2014), pp. 198-213), a high-order entropy viscosity method, and the Boris-Book-Zalesak flux correction technique. The algorithm works for arbitrary meshes in any space dimension and for all Lipschitz fluxes. The formal second-order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems.

Adaptive simulation of turbulent flow past a full car model

The massive computational cost for resolving all turbulent scales makes a direct numerical simulation of the underlying Navier-Stokes equations impossible in most engineering applications. We present recent advances in parallel adaptive finite element methodology that enable us to efficiently compute time resolved approximations for complex geometries with error control. In this paper we present a LES simulation of turbulent flow past a full car model, where we adaptively refine the unstructured mesh to minimize the error in drag prediction. The simulation was partly carried out on the new Cray XE6 at PDC/KTH where the solver shows near optimal strong and weak scaling for the entire adaptive process.

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

The high order spectral element approximation of the Euler equations is stabilized via a dynamic sub-grid scale model (Dyn-SGS). This model was originally designed for linear finite elements to solve compressible flows at large Mach numbers. We extend its application to high-order spectral elements to solve the Euler equations of low Mach number stratified flows. The major justification of this work is twofold: stabilization and large eddy simulation are achieved via one scheme only.Because the diffusion coefficients of the regularization stresses obtained via Dyn-SGS are residual-based, the effect of the artificial diffusion is minimal in the regions where the solution is smooth. The direct consequence is that the nominal convergence rate of the high-order solution of smooth problems is not degraded. To our knowledge, this is the first application in atmospheric modeling of a spectral element model stabilized by an eddy viscosity scheme that, by construction, may fulfill stabilization requirements, can model turbulence via LES, and is completely free of a user-tunable parameter.From its derivation, it will be immediately clear that Dyn-SGS is independent of the numerical method; it could be implemented in a discontinuous Galerkin, finite volume, or other environments alike. Preliminary discontinuous Galerkin results are reported as well. The straightforward extension to non-linear scalar problems is also described. A suite of 1D, 2D, and 3D test cases is used to assess the method, with some comparison against the results obtained with the most known Lilly-Smagorinsky SGS model.

Unicorn: a unified continuum mechanics solver

This chapter provides a description of the technology of Unicorn focusing on simple, efficient and 10597 general algorithms and software for the Unified Continuum (UC) concept and the adaptive General 10598 Galerkin (G2) discretization as a unified approach to continuum mechanics.

Convergence of a residual based artificial viscosity finite element method

We present a residual based artificial viscosity finite element method to solve conservation laws. The Galerkin approximation is stabilized by only residual based artificial viscosity, without any least-squares, SUPG, or streamline diffusion terms. We prove convergence of the method, applied to a scalar conservation law in two space dimensions, toward an unique entropy solution for implicit time stepping schemes.

On the Stability of the Dual Problem for High Reynolds Number Flow Past a Circular Cylinder in Two Dimensions

In this paper we present a computational study of the stability of time dependent dual problems for compressible flow at high Reynolds numbers in two dimensions. The dual problem measures the sensitivity of an output functional with respect to numerical errors and is a key part of goal oriented a posteriori error estimation. Our investigation shows that the dual problem associated with the computation of the drag force for the compressible Euler/Navier--Stokes equations, which are approximated numerically using different temporal discretization and stabilization techniques, is unstable and exhibits blow-up for several Mach regimes considered in this paper.

Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.

Goal-oriented adaptive finite element methods for elliptic problems revisited

A goal-oriented a posteriori error estimation of an output functional for elliptic problems is presented. Continuous finite element approximations are used in quadrilateral and triangular meshes. The algorithm is similar to the classical dual-weighted error estimation, however the dual weight contains solutions of the proposed patch problems. The patch problems are introduced to apply Clement and Scott-Zhang type interpolation operators to estimate point values with the finite element polynomials. The algorithm is shown to be reliable, efficient and convergent.