**Publications**

**Space-Time Discontinuous Galerkin Cell-Vertex Scheme (DG-CVS)**(new)

__S. Tu__, “A Riemann-solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws,”*American Journal of Computational Mathematics*, Vol. 5, No. 2, (2015), pp. 55-74.__S. Tu__and H. Song and L. Ji and Q. Pang, “Further Development of a Riemann-solver Free Space-time Discontinuous Galerkin Method for Compressible Magnetohydrodynamics (MHD) Equations”, presented at AIAA Orlando SciTech Conference 2015, AIAA Paper 2015-0567.__S. Tu__, Q. Pang and R.S. Myong, "A Riemann-solver Free Space-time Discontinuous Galerkin Method for Magnetohydrodynamics," AIAA Paper 2013-2755. presented at the 44th AIAA Plasmadynamics and Lasers Conference, June 24-27, 2013, San Diego, CA.__S. Tu__and Q. Pang, “Development of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) for Moving Mesh Problems,” AIAA Paper 2012-2835. presented at the 2012 AIAA Fluid Dynamics Conference in New Orleans, LA.__S. Tu__, Q. Pang and H. Xiang, "Solving the Shallow Water Equations Using the High Order Space-time Discontinuous Galerkin Cell Vertex Scheme." AIAA Paper 2012-0307. Presented at the 50th AIAA Aerospace Science Meeting, Jan. 9-12, 2012, Nashville, TN.__S. Tu__, Q. Pang and H. Xiang, "Solving the Level Set Equation Using the High Order Space-time Discontinuous Galerkin Cell Vertex Scheme." AIAA Paper 2012-1233. Presented at the 50th AIAA Aerospace Science Meeting, Jan. 9-12, 2012, Nashville, TN.__S. Tu,__G.W. Skelton and Q. Pang, “Extension of the High-Order Space-Time discontinuous Galerkin Cell Vertex Scheme for Time Dependent Diffusion Equations,”*Communications in Computational Physics*, (2012), Vol. 11, No. 5, pp 1503-1524.

**Abstract:**In this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods. The numerical tests demonstrate that the convergence order based on the L_{2} -norm is optimal, i.e. \mathcal{O}(h^{p+1}) for the solution and \mathcal{O}(h^{p}) for the solution gradients, when the basis polynomials are of odd degrees. For even-degree polynomials, the convergence order is sub-optimal for the solution and optimal for the solution gradients. The same odd-even behaviour can also be seen in some other DG-typed methods.- S. Tu, Q. Pang and H. Xiang, "Wave Computation Using a A High Order Space-time Riemann Solver Free Method." AIAA Paper 2011-2846. Presented at the 17th AIAA/CEAS Aeroacoustics Conference, June 5-8, 2011, Portland, Oregon, USA.
__S. Tu__, G.W. Skelton, and Q. Pang, “Solving Time Dependent Diffusion Equations on Arbitrary Grids Using the High-Order Space-Time discontinuous Galerkin Cell Vertex Scheme,” AIAA Paper 2011-0050. Presented at the 49th AIAA Aerospace Science Meeting, Jan. 4-7, 2011, Orlando, FL.

Abstract:In this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. The treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are continuous across the cell interface in DG-CVS. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional spatial DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods. Some numerical results are presented to demonstrate the accuracy of the method.

__S. Tu__, G.W. Skelton and Q. Pang, “A compact high order space-time method for conservation laws.”*Communications in Computational Physics*, (2011), Vol. 9, No. 2, pp 441-480.

Abstract:This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

__S. Tu__and Z. Tian, “Preliminary implementation of a high order space-time method on overset Cartesian/quadrilateral grids,” AIAA Paper 2010-0544. Presented at the 48th AIAA Aerospace Science Meeting, Jan. 4-7, 2010, Orlando, FL.

Abstract:This paper reports our current status in implementing our high-order space-time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) for conservation laws on overset Cartesian/quadrilateral grids. The motivation to do this is to improve the efficiency of the DG-CVS and overcome the difficulty in generating meshes around complex geometries. The preliminary results of the inviscid flow passing a circular cylinder will show some promises and challenges of the current implementation.

__S. Tu__, “A solution limiting procedure for a high order space-time method,” AIAA Paper 2009-3983. Presented at the 19th AIAA Computational Fluid Dynamics Conference, Jun. 22-25, 2009, San Antonio, TX.

Abstract:In this paper, a solution limiting procedure is described for a high-order space-time method for conservation laws. The underlying high-order method is a space-time discontinuous Galerkin cell-vertex scheme (DG-CVS) for solving hyperbolic differential equations. The proposed limiting procedure is activated only in truly oscillatory regions detected by a reliable oscillation indicator. The limiter limits the averaged solution derivatives while preserving the cell average. Therefore, the limiter is conservative. The limiter ensures the solution satisfies the following two constraints: (i) the solution does not exceed the maximum or minimum cell averages in the local stencil; and (ii) the solution gradient is consistent with the local solution variation across each edge. The limiting procedure is recast as a quadratic programming problem with linear inequality constraints, which is solved by the active set method. It is hoped that through the quadratic programming, the optimum values of limiting factors can be obtained while satisfying the constraints. For arbitrary high order solutions, these constraints are formulated as the sufficient (not necessary) conditions. The constraints are enforced along the directions connecting adjacent two nodes, thus the method is suitable for arbitrarily unstructured spatial meshes. Several 1D and 2D examples will demonstrate that the limiter is able to effectively suppress the unphysical oscillations around discontinuities while preserving the high-order accuracy in smooth regions as well as the local extrema.

__S. Tu__, “A high order space-time Riemann-solver-free method for solving compressible Euler equations,” AIAA Paper 2009-1335. Presented at the 47th AIAA Aerospace Science Meeting in Orlando, Florida, Jan. 5-8, 2009.

Abstract:In this paper, a novel high-order space-time method is introduced for solving compressible Euler equations. The method is inspired by two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The solution within each physical time step is updated alternately at the cell level and the vertex level. The arbitrarily high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each CE. The current method exhibits many advantageous features such as Riemann-solver-free, arbitrarily high-order accuracy, point-implicitness, and compactness. Several numerical tests will demonstrate the performance of the new scheme.

**Hybrid Finite Volume/Element Method**

__S. Tu__, S. Aliabadi, M. Watts and R. Patel, “An Implementation of the Spalart-Allmaras DES Model in an Implicit Unstructured Hybrid Finite Volume/Element Solver for Incompressible Turbulent Flow,”*International Journal of Numerical Methods in Fluids*, Vol. 59, No. 9, (2008) pp. 1051-1062.

In this paper, we extend our implicit hybrid finite volume/element incompressible Navier-Stokes solver to turbulent flows by adding the Spalart-Allmaras Detached Eddy Simulation (SA-DES) turbulence equation to the governing equation set. The baseline hybrid solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix-free implicit cell-centered finite volume method. The pressure Poisson equation is solved by the node-based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered-mesh scheme. The SA-DES turbulence equation is solved after the velocity field and the pressure field have been updated at the end of each timestep. The same matrix-free finite volume method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA-DES model in a hybrid baseline flow solver. This paper will address important implementation issues for high-Reynolds number flows where highly stretched elements are typically used. Also, some aspects of implementing the SA-DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flat plate flow and a high-Reynolds number flow around a high angle-of-attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation.Abstract:

__S. Tu__and S. Aliabadi, “Development of a Hybrid Finite Volume/Element Solver for Incompressible Flows on Unstructured Meshes,”*International Journal of Numerical Methods in Fluids*, Vol. 55, No. 2, (2007) pp. 177-203.

Abstract:In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier-Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix-free implicit cell-centered finite volume method. The Poisson equation derived from the fractional step approach is solved by the node-based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at cell vertices, making the current solver a staggered-mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady-state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder.

**Parallel and Vectorized Finite Volume Solver**

- M. Watts,
__S. Tu__and S. Aliabadi, “Numerical Simulation of a Spinning Projectile Using Parallel and Vectorized Unstructured Flow Solver,” Parallel Computational Fluid Dynamics 2007. Also appears in*Lecture Notes in Computational Science and Engineering*, Vol. 67, pp. 1-8, 2009. DOI: 10.1007/978-3-540-92744-0_1. __S. Tu__, M. Watts, A. Fuller, R. Patel and S. Aliabadi, “Development and Performance of CaMEL_Aero, a Truly Matrix-free, Parallel and Vectorized Unstructured Finite Volume Solver for Compressible Flows,” in*Proceedings of 25th Army Science Conference*, Nov. 2006, Orlando, FL.

Abstract:This paper reports the development and performance of CaMEL_Aero, our truly matrix-free, parallel and vectorized unstructured finite volume solver for compressible flows. The Jacobian-free GMRES method is used to solve the linear systems of equations inside each nonlinear Newton-Raphson iteration. Furthermore, the matrix-free Lower-Upper Symmetric Gauss Seidel (LU-SGS) method is employed as a preconditioning technique to the GMRES solver. The solver is parallelized using mesh partitioning and Message Passing Interface (MPI) functions. The solver is also vectorized using two main vectorization techniques: the face coloring algorithm to vectorize the long loops over faces and the truncated Neumann expansions of the inverse of preconditioning matrices to vectorize the LU-SGS preconditioner, respectively. A few 2D and 3D numerical examples are presented to demonstrate the performance of the present solver.

__S. Tu__and S. Aliabadi, A. Johnson and M. Watts, “High Performance Computation of Compressible Flows on the Cray X1,” in*Proceedings of the Second International Conference on Computational Ballistics*, pp. 347-356, on May 18 - 20, 2005, Cordoba, Spain.__S. Tu__, S. Aliabadi, A. Johnson and M. Watts, “A Robust Parallel Implicit Finite Volume Solver for High-speed Compressible Flows,” AIAA paper 2005-1396. Presented at the 43rd AIAA Aerospace Sciences Meeting and Exhibit on January 10-13, 2005, Reno, Nevada.- S. Aliabadi,
__S. Tu__, M.D. Watts, “High Performance Computing of Compressible Flows,” hpcasia, pp. 153-160, in*Proceedings of the Eighth International Conference on High-Performance Computing in Asia-Pacific Region*(HPCASIA'05), 2005

**Space-Time Cell-Vertex Scheme (CVS)**

__S. Tu__and S. Aliabadi, “A space-time upwind cell-vertex scheme for conservation laws: a Riemann solver-free approach,” in the*Proceedings of the third M.I.T. Conference on Computational Fluid and Solid Mechanics*, pp. 1191-1195, Cambridge, MA, 2005.

**Discontinuous Galerkin Solver**

__S. Tu__and S. Aliabadi, “A Slope Limiting Procedure in Discontinuous Galerkin Finite Element Method for Gasdynamics Applications,”*International Journal of Numerical Analysis and Modeling*, Vol. 2, No. 2, (2005) pp. 163-178.__S. Tu__and S. Aliabadi, “Computation of High Temperature Equilibrium Airflows Using Discontinuous Galerkin Finite Element Method,”*Communications in Numerical Methods in Engineering*, Vol. 21, No. 12, (2005) pp. 735-745.- S. Aliabadi and
__S. Tu__, "An Alternative to Limiter in Discontinuous Galerkin Finite Element Method,” AIAA paper 2004-0076. Presented at the 42nd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 2004.

**Solution Adaptive Cartesian-Grid Solver**

__S. Tu__and S.M. Ruffin, “Solution Adaptive, Unstructured Cartesian-Grid Methodology for Chemically Reacting Flow,” AIAA paper 2002-3097. Presented at the 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, St. Louis, June 2002.__S. Tu__and S.M. Ruffin, “Calculation of Non-equilibrium Flows Using a Solution Adaptive, Unstructured Cartesian-Grid Solver,” AIAA paper 2002-3098. Presented at the 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, St. Louis, June 2002.

**Parallel Mesh Subdivision**

__S. Tu__, S. Aliabadi, and M. Watts, “A Parallel Program for Subdividing 3-D Hybrid Unstructured Meshes,” in*Proceedings of 10th ISGG Conference on Numerical Grid Generation*, Sept. 16-20, 2007, Forth, Crete, Greece.

**Stabilized Continuous Finite Element Solver**

- S. Aliabadi,
__S. Tu__, M. Watts, A. Ji and A. Johnson, “Integrated High Performance Computational Tools for Simulations of Transport and Diffusion of Contaminants in Urban Areas,”*International Journal of CFD*, Vol. 20, No. 3-4, (2006) pp. 253-267. - S. Aliabadi, A. Johnson, J. Abedi,
__S. Tu__and A. Tate, “High Performance Simulation of Contaminant Dispersion on the Cray X1: Verification and Implementation,”*Journal of Aerospace Computing, Information, and Communication*, Vol. 1, No. 8, (2004) pp. 341-361. - M. Watts, S. Aliabadi,
__S. Tu__and C. Bigler, “Free-Surface Flow Computations Using Scalable Parallel Finite Element Method,” in Proceedings of Parallel CFD Conference, May 21-24, 2007, Antalya, Turkey. - M. Watts, S. Aliabadi and
__S. Tu__, “A Coupled Interface-Tracking/ Interface-Capturing Technique for Free-surface Flows,” in*Proceedings of the Third International Conference on Computational Methods in Multiphase Flow*, pp. 353-363, Oct. 30 . Nov. 2 2005, Portland, Maine. - S. Aliabadi and
__S. Tu__, “Simulation of Cavitating Flows Using Stabilized Finite Element Method,” AIAA paper 2005-1288. Presented at the 43rd AIAA Aerospace Sciences Meeting and Exhibit held on 10-13 January 2005 at the Reno Hilton in Reno, Nevada.