• Abstract

This work presents two computational efficiency improvements for the hybridizable discontinuous Galerkin (HDG) fluid-structure interaction (FSI) model presented by Sheldon et al. A new formulation for the solid is presented that eliminates the global displacement, resulting in the velocity being the only global solid variable. This necessitates a change to the solid-mesh displacement coupling, which is accounted for by coupling the local solid displacement to the global mesh displacement. Additionally, the mesh basis and test functions are restricted to linear polynomials, rather than being equal-order with the fluid and solid. This change increases the computational efficiency dynamically, with greater benefit the higher order the computation, when compared to an equal-order formulation. These two improvements result in a 50% reduction in the number of global degrees of freedom for high-order simulations for both the fluid and solid domains, as well as an approximately 50% reduction in the number of local fluid domain degrees of freedom for high-order simulations. The new, more efficient formulation is compared against that from Sheldon et al. and negligible change of accuracy is found.

• Abstract

A direct Arbitrary Lagrangian Eulerian (ALE) method based on multi-moment finite volume scheme is developed for the Euler equations of compressible gas in 1D and 2D space. Both the volume integrated average (VIA) and the point values (PV) at cell vertices, which are used for high-order reconstructions, are treated as the computational variables and updated simultaneously by numerical formulations in integral and differential forms respectively. The VIAs of the conservative variables are solved by a finite volume method in the integral form of the governing equations to ensure the numerical conservativeness; whereas, the governing equations of differential form are solved for the PVs of the primitive variables to avoid the additional source terms generated from moving mesh, which largely simplifies the solution procedure. Numerical tests in both 1D and 2D are presented to demonstrate the performance of the proposed ALE scheme. The present multi-moment finite volume formulation consistent with moving meshes provides a high-order and efficient ALE computational model for compressible flows.

• Abstract

This work is devoted to the derivation of an admissible asymptotic-preserving scheme for the electronic $M_1$ model in the diffusive regime. A numerical scheme is proposed in order to deal with the mixed derivatives which arise in the diffusive limit leading to an anisotropic diffusion. The derived numerical scheme preserves the realisability domain and enjoys asymptotic-preserving properties correctly handling the diffusive limit recovering the relevant limit equation. In addition, the cases of non constants electric field and collisional parameter are naturally taken into account with the present approach. Numerical test cases validate the considered scheme in the non-collisional and diffusive limits.

• Abstract

Residual-based posteriori error estimates for discontinuous streamline diffusion methods for transport equations are studied in this paper. Computable upper bounds of the errors are measured based on mesh-dependent energy norm and negative norm. The estimates obtained are locally efficient, and thus suitable for adaptive mesh refinement applications. Numerical experiments are provided to illustrate underlying features of the estimators.

• Abstract

In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.

• Abstract

We have recently proposed a new meshless method for solving second order partial differential equations where the polynomial particular solutions are obtained analytically [1]. In this paper, we further extend this new method for the solution of general two- and three-dimensional Cauchy problems. The resulting system of linear equations is ill-conditioned, and therefore, the solution will be regularized by using a multiple scale technique in conjunction with the Tikhonov regularization method, while the L-curve approach is used for the determination of a suitable regularization parameter. Numerical examples including 2D and 3D problems in both smooth and piecewise smooth geometries are given to demonstrate the validity and applicability of the new approach.

• Abstract

This paper mainly focuses on an efficient numerical method for the optimization problem constrained with a stationary Maxwell system. Following the idea of [32], the edge element is applied to approximate the state variable and the control variable, then the continuous optimal control problem is discretized into a finite dimensional one. The novelty of this paper is the approach for solving the discretized system. Based on the separable structure, an alternating direction method of multipliers (ADMM) is proposed. Furthermore, the global convergence analysis is established in the form of the objective function error, which includes the discretization error by the edge element and the iterative error by ADMM. Finally, numerical simulations are presented to demonstrate the efficiency of the proposed algorithm.

• Abstract

In this paper, we present a new monotone finite volume scheme for the steady state convection-diffusion equation. The discretization of diffusive flux [33] is utilised and a new corrected upwind scheme with second order accuracy for the discretization of convective flux is proposed based on some available informations of diffusive flux. The scheme is locally conservative and monotone on deformed meshes, and has only cell-centered unknowns. Numerical results are presented to show that the scheme obtains second-order accuracy for the solution and first-order accuracy for the flux.

• Abstract

A state-constrained optimal control problem governed by the stationary Navier-Stokes equations is studied. Finite element approximation is constructed, the optimal-order a priori H¹-norm and L²-norm error estimates are given, for which the optimal state is a nonsingular solution of the Navier-Stokes equations to the optimal control.

• Abstract

Accurate numerical simulation of fluid flows around wind turbine plays an important role in understanding the performance and also the design of the wind turbine. The computation is challenging because of the large size of the blades, the large computational mesh, the moving geometry, and the high Reynolds number. In this paper, we develop a highly parallel numerical algorithm for the simulation of fluid flows passing three-dimensional full size wind turbine including the rotor, nacelle, and tower with realistic geometry and Reynolds number. The flow in the moving domain is modeled by unsteady incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form and a non-overlapping sliding-interface method is used to handle the relative motion of the rotor and the tower. A stabilized moving mesh finite element method is introduced to discretize the problem in space, and a fully implicit scheme is used to discretize the temporal variable. A parallel Newton-Krylov method with a new domain decomposition type preconditioner, which combines a non-overlapping method across the rotating interface and an overlapping Schwarz method in the remaining subdomains, is applied to solve the fully coupled nonlinear algebraic system at each time step. To understand the efficiency of the algorithm, we test the algorithm on a supercomputer for the simulation of a realistic 5MW wind turbine. The numerical results show that the newly developed algorithm is scalable with over 8000 processor cores for problems with tens of millions of unknowns.

• Abstract

This paper contributes to apply both the direct Eulerian and Lagrangian generalized Riemann problem (GRP) schemes for the simulation of compressible fluid flows in two-dimensional cylindrical geometry. Particular attention is paid to the treatment of numerical boundary conditions at the symmetric center besides the zero velocity (momentum) enforced by the symmetry. The new treatment precisely describes how the thermodynamical variables are discretized near the center using the conservation property. Moreover, the Lagrangian GRP scheme is verified rigorously to satisfy the properties of symmetry and conservation. Numerical results demonstrate the performance of such treatments and the symmetry preserving property of the scheme with second order accuracy both in space and time.

• Abstract

This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [5] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.