N C H C 96 / 12 / 20 T.-I Tseng T.-I Tseng National Center for High-performance Computing The Development and Application of the Space-Time Conservation Element and Solution Element Method Institute of Physics, NCTU Dec 20, 2007

2 Background of the Space-Time CE/SE Method

3 Transport Theory  Mass conservation in continuum mechanics  Transport theory x y z

4 The Finite Volume Method  The rate of change of conserved properties in a finite volume (FV) is equal to its flux across cell boundaries  The FV methods focus on calculating spatial flux (a temporal evolving spatial flux). x t FV

5 The Space-Time CE/SE Method  The convection equation is a space- time divergence free condition  Let x 1 = x and x 2 = t be the coordinates of a 2D Euclidean space E 2  Using Gauss’ divergence theorem, one obtains the space- time flux balance equation

6 The Space-Time CE/SE Method  The Euler equations are three space-time divergence free conditions  Using Gauss’ divergence theorem, one obtains the space- time flux balance equation

7 Space-Time CE/SE Method A staggered space-time mesh  the discretization step  Space time region is divide into non- overlapping Conservation Elements (CEs) and Solution Elements (SEs).

8 The Solution Element  Flow properties are assumed continuous inside each SE.  The 1st order Taylor series expansion is used inside a SE  Inside a SE,  U and U x are the unknowns to be solved; all other properties can be expressed by them. (j,n) (j-1/2,n-1/2)(j+1/2,n-1/2) (j,n) Solution element; SE(j, n)

9 The Conservation Element  The space-time region is divided into non-overlapping CEs  The space-time flux conservation is imposed over CE - and CE +  For one conservation equation, CE - and CE + provide two conditions.  For the 1D Euler equations, CE - and CE + provide 6 conditions for the 6 components of U and U x at point (j, n) CE-CE+ (j,n) (j-1/2,n-1/2)(j+1/2,n-1/2) x t B A F C D E (j,n) (j-1/2,n-1/2) (j+1/2,n-1/2) (j-1/2,n-1/2) (j,n) B A C D (j,n) (j+1/2,n-1/2) A F D E Conservation element; CE(j, n) Basic Conservation element; BCE(j, n) CE - CE +

10 2D Space-Time Mesh  Triangular unstructured mesh  Quad cylinders for CEs  3 CEs between A, E, C and G’, for the 3 unknowns U, U x and U y. G A B C D E F x y x y t A C E G G’ Time marching

11 The CE and SE in 2D  Three CEs: Quadrilateral cylinder EFGDEFGD(1) CDGBCDGB(2), and ABGFABGF ( 3 ).  One SE: Four planes ABCDEF + GG  B  B + GG  D  D + GG  F  F + their immediate neighborhood.

12 3D Space-Time Mesh  Tetrahedrons are used as the basic shape  Every mesh node has 4 neighboring nodes  The projection of a space- time CE on the 3D space is a 6-surface polygons  Flux conservation over 4 CEs determine the 4 unknowns: U, U x, U y, and U z x z y

13 Special Features of the CE/SE Method  Space and time are unified and treated as a single entity.  Separation of conservation element and solution element.  No flux function or characteristics-based techniques and no reconstruction step.  Numerical dissipation doesn’t overwhelm physical dissipation  Use the simplest mesh stencil -- triangles for 2D and tetrahedrons for 3D.  1,2, and 3 D Euler/NS codes for structured/unstructured meshes running on serial and parallel platforms.  Many application in aero acoustics and combustion.

14 A Space-Time CE/SE Method with Moving Mesh Scheme for One-Dimensional Hyperbolic Conservation Laws

15 Motivating idea  A suitable computational grid is important for solving the hyperbolic systems.  Major challenge of numerical scheme is to capture the discontinuous solution with sufficient accuracy.  The characteristic of the discontinuous is non- stationary and consequently the fixed uniform grid may not be the best suited.  The idea of an adaptive grid is to add, remove, or move the grid concentrated to enhance accurate and achieve efficiency.

16 Adaptive Mesh  Adaptive Mesh Refinement (AMF): automatic refinement or coarsening of the spatial mesh.  Adaptive Mesh Redistribution (AMF): relocates the grid points with a fixed number of nodes. it’s also known as moving mesh method (MMM).  Key ingredients of the moving mesh method include:  Mesh equation  Monitor function  Interpolations n n’ n+1/2  Interpolation Free MMM : Interpolation of dependent variables from the old mesh to the new mesh is unnecessary.

17 Mesh Equation The logical and physical coordinates: Equidistribution principles x :  : (Quasi-Static equidistribution principles;QSEPs)

18 Monitor Function Scaled solution arc-length where  is a scaling parameter and the cell average of the solution gradient over the interval [x j, x j+1 ]  = 0, uniform mesh  >> 1, adapted grid Smoothing the monitor function

19 MMCESE Moving mesh strategy: use the Gauss-Seidel iteration to solve the mesh equation Moving mesh CE/SE (MMCESE)method

20 MMCESE Algorithm  Step 1: Given a uniform partition of logical and physical domains, then specified the initial conditions.  Step 2 : Calculate the monitor function.  Step 3 : Move the grid point by mesh equation.  Step 4 : Evolve the underlying PDEs by CE/SE method on the new mesh system to obtain the flow variables at new time level.  Step 5 : If t n+1 < T, go to Step 2.

21 Burger Equation Initial condition : Monitor function :

22 Sod Problem Initial conditions : Monitor function :

23 Piston Problem Moving Boundary

24 Conclusion  MMCESE not only maintains the essential features of the original CE/SE method but also clusters the mesh space at the locations where large variation in physical quantities exists.  Current approach can be extend to moving boundary problem easily.  MMCESE is an interpolation free MMM.  Computation accuracy and efficiency can be improved by this approach. Tseng, T.I, and Yang R.J.. “A Space-Time Conservation Element and Solution Element Method with Moving Mesh Scheme for One- Dimensional Hyperbolic Conservation Laws,” the 6 th Asian Computational Fluid Dynamics Conference, 2005.

25 Applications in Shallow Water Equations

26 Shallow Water Equations Depth averaging of the free surface flow equations under the shallow-water hypothesis leads to a common version of the shallow-water equations (SWEs) where s 0 and s f are bed slop and friction slop, respectively. The friction slop is determined by the Manning formula n is the Manning roughness coefficient.

27 CE/SE method with Source terms By using Gauss’ divergence theorem in space-time region, the SWEs can be written in integral form For any point belong the solution element From SWEs, one can get As a result, there are two independent marching variables and associated with in each solution elements. Furthermore

28 CE/SE method with Source terms We employ local space-time flux balance over conservation element to solve the unknowns, i.e., because the boundary of CE(j, n) is a subset of the union of SE(j, n), SE(j-1/2, t-1/2), and SE(j+1/2, t-1/2), the conservation laws imply that The space derivatives of flow variables are evaluated using the α scheme. Yu, S.T., and Chang, S.C. “Treatments of Stiff Source Terms in Conservation Law by the Method of Space-Time Conservation Element and Solution Element,”, AIAA Paper ,1997 B A F C D E (j,n) (j-1/2,n-1/2) (j+1/2,n-1/2)

29 1D Dam-Break Problem with Finite Downstream Water Depth Two different initial water depths are assigned to the upstream and downstream parts of a horizontal, frictionless, infinitely wide rectangular channel including a dam. The upstream water depth is 100 m and the downstream one is 1 m. Spatial domain is 2000 m length and it is discretized with 200 elements.

30 Propagation and Reflection It is assumed that a dam located in the middle of a 1000 m long channel separates reservoir with depth h r and tailwater depth h t. At the ends of the channel are the walls that the wave cannot surmount. The boundary conditions with free-slip and zero discharge are satisfied at the ends of the channel. At time t = 30 s and t = 75 s after dame break, the results with different ratios of water depth (R = 0.15 and 0.001)

31 Propagation and Reflection It is assumed that a dam located in the middle of a 1000 m long channel separates reservoir with depth h r and tailwater depth h t. Comparisons the numerical results with and without the bed friction at time t = 30 s after dame with bed slop and different ratios of water depth (R = 0.15 and 0.001).

32 Interaction of 1D Bore Waves It is supposed that in two 1000 m long channels there are two dams located at 300 m and 600 m, respectively. Each channel is divided into three parts by both dames. The initial clam water depths are h 01, h 02, and h 03. (1) h 01 = 20 m, h 02 = 3 m, and h 03 = 10 m; (2) h 01 = 20 m, h 02 = 10 m, and h 03 = 2 m;

33 Steady flow over a bump with hydraulic jump A steady-state transcritical flow over a bump, with a smooth transition followed by a hydraulic jump is simulated. The channel is infinitely large, horizontal, frictionless, 25 m long.

34 2D Dam-Break Problem A square box of 200 ╳ 200 m 2 with a horizontal bed is divided into two equal compartments. The initial still water depth is 10 m on one side and 5m and 0.01m on the other side of the dividing wall for the wet bed and dry test cases, respectively. The breach is 75 m in length, and the dame is 15 m in thickness. Tseng, T.I, and Yang R.J.. “Solution of Shallow Water Equations Using Space-Time Conservation Element and Solution Element Method,” the 14 th National Computational Fluid Dynamics Conference, 2007.

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36 Acknowledgement Dr. S.-C., Chang NASA Gleen Research Center Prof. S.-T., Yu Ohio State University Dr. C.-L., Chang NASA Langley Research Center Prof. R.-J., Yang NCKU Dr. Z.-C., Zhang Livermore Software Technology Co. Prof. W.-Y., Sun NCTFR

37 Thanks for Your Attention