AMR Multi-Moment FVM Scheme

Chapter 4 AMR Multi-Moment FVM Scheme 4.1 Berger s AMR scheme An AMR grid with the hierarchy of Berger s AMR scheme proposed in [13] for CFD simulations is given in Fig.4.1 as a simple example for following discussion. Different from their original work, we don t consider the rotated grid in this study. The base grid, i.e. B 1,1, covers the whole computational domain using a resolution defined by user in advance and remains fixed during the computation. For sake of simplicity, the computational domain is rectangular and these is only one base grid for the discussion here. In practical applications, it maybe contains more than one base grids, and the computational domain isn t necessary to be rectangular if we use the general curvilinear coordinates, for example on cubed-sphere grid. Three nesting levels are involved in this example AMR grid. Two blocks have resolution of level 2 and one block has the finest resolution of level 3. Suppose the uniform grid spacing of h 1, h 2 and h 3 adopted for blocks on nesting level 1, 2 and 3, the refinement ratio h i h i 1 in Berger s scheme can be any integer larger than 1. Two important properties of this AMR grid should be pointed out. They are 139

14 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME The blocks except those on the base level overlay the blocks on coarser levels to assure local uniformity of each block. And each block can be independently integrated during the computations; The fine blocks should be contained by those on the next coarse level although it isn t necessary to be surrounded by one block, for example the block B 3,1 in Fig.4.1. In other words, all discrete points of the fine blocks should be the internal points of the area covered by blocks on next coarse level except on the physical boundaries. Figure 4.1: An example AMR grid with the hierarchy of Berger s scheme. 4.1.1 Data structure for AMR grid Each block is defined independently and has its own storage space for flow solution. Dynamic allocation of the memory and using user-defined variable type are provided in Fortran 9 as the new features compared with Fortran 77. A new variable type, namely block, is designed in our code, the information that should be stored includes [13, 2]

4.1. BERGER S AMR SCHEME 141 location and size of the block; the nesting level; present time instant; solution arrays and auxiliary arrays necessary during solution procedure; the pointer to its next sibling block; the values of the fluxes across boundary between coarse and fine nesting levels for the conservation corrections. A Fortran 9 code can be used for defining new variable type of block as MODULE defblock TYPE block! index of left-bottom corner INTEGER nxlo,nylo! size of the block INTEGER nx,ny! nesting level INTEGER lev! present time instant REAL*8 time! solution arrays for two dimensional systems REAL*8, DIMENSION(:,:,:), ALLOCATABLE :: qold,qnew

142 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME! link to sibling block TYPE(block), POINTER :: nextblock! flux for conservation corrections REAL, DIMENSION(:,:) :: cflux,fflux! indexes of neighboring elements on next coarse level INTEGER, DIMENSION(:,:) :: cne END TYPE block END MODULE defblock Besides the variables defined above, there are also some necessary auxiliary arrays defined in our code which are not mentioned above for sake of brevity. To link the blocks on different nesting levels, a pointer array lvst is defined as TYPE(block), POINTER :: lvst(:) Each member of array lvst points to the first block on each nesting level. As a result, the solution of the physical field for whole computational domain is stored in a number of variables of new-defined type block and all blocks are connected together by pointers also of type block. For the above example grid, the structure of data storage space is illustrated by Fig.4.2. In the following part of this section, we briefly review the Berger s AMR scheme. The more detailed description can be found in [13, 12] and open source AMR library AMRCLAW [2].

4.1. BERGER S AMR SCHEME 143 Figure 4.2: The data structure for example AMR grid shown in Fig.4.1. 4.1.2 Generating the AMR grid Initiating AMR grid We begin the description of generating the AMR grid from the initial setup of the AMR grid. The task of generating blocks on some nesting level can be described as follows. The variable lev denotes nesting level, lev [1, maxlv] where maxlv is the maximal available nesting level determined by user. Without losing generality, we consider a square computational domain in Cartesian coordinates, and D is length of side. The grid spacing on nesting level lev is h lev if an uniform grid is constructed. The total number of computational elements are n lev = D h lev in both x and y directions. The control volume on the nesting level lev is represented by C lev i j where subscripts i, j [1, n lev ]. Donation Ω is used here to mean a set of computational elements. Additionally, we use B lev i to mean the i th blocks on nesting level lev where i [, m lev ] and m lev is total number of blocks on nesting level lev. The denotations are shown in Fig.4.3. Provided the blocks on nesting level k have been determined and m k >, we

144 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME proceed to generate the blocks on the nesting level k + 1 if k + 1 maxlv, the procedure can be described as follows. Figure 4.3: Denotations for describing the generation of AMR grid. Step one: preparing the set of elements available for refinement The fist step to generate the new blocks is to determine the elements which are allowed to be refined by the nesting relation from level k. We specify the set Ω c as Ω c = { C k i j : if f i j = 1 }, (4.1) where function f i j is specified by 1, if B k p%nxlo i B k p%nxlo + nx 1 and f i j = B k p%nylo j B k p%nylo + ny 1, otherwise, (4.2) 1 p m k and using symbol % follows the syntax of Fortran language. Ω a, the set of elements allowed to be refined by nesting relation, is a subset of Ω c, Which is determined by the requirement that all elements involved in new level should be the internal elements of the next coarse level. It is expressed as Ω a = { C k i j : if f i j = 1 and f i±1 j±1 = 1, f i j±1 = 1, f i±1 j = 1 }. (4.3)

4.1. BERGER S AMR SCHEME 145 Obviously, to determine the set Ω a, the values of function f for one-layer ghost cells should be determined. On the boundary of the computational domain, the value of function f is determined by considering the different boundary conditions. For those boundaries which are periodic or shared with another computational domain, e.g. patch boundaries on cubed sphere, status of ghost cells can be determined by considering the Ω c periodically or from the Ω c in neighboring computational domain. For those boundaries where the boundary condition is supplied by using some approximate formulations, the value of function f is determined by { f, j = f 1, j, f nk +1, j = f nk, j ( j = 1, n k ) for left and right edges f i, = f i,1, f i,nk +1 = f i,nk (i = 1, n k ) for top and bottom edges (4.4) along four boundary edges and f, = f,1 f 1,, f,nk +1 = f,nk f 1,nk +1, f nk +1, = f nk, f nk +1,1, f nk +1,n k +1 = f nk,n k +1 f nk +1,n k (4.5) at four vertices of enlarged computational domain. Step two: flagging the elements to be refined The elements which are necessary to be refined are determined by using refinement criterion RC. The set Ω n consisting of elements satisfying RC is determined Ω n = { C k i j : if RC is true}. (4.6) Generally, the existing criteria can be classified into two types. The first type is based on truncation error estimations, e.g. Richardson-type criterion used in original Berger s scheme [13]. The second type consists of those methods based on the direct analysis of the physical field, for example gradient-based criterion

146 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME often used for discontinuous fields or vorticity-based one in some atmospheric simulations. Some applications and analysis of the different refinement criteria can be referred to [8, 26, 31]. The methods of second type are usually easier to be implemented with less computational overhead compared with those of type one, but they are problem-dependent. In present study, we use the criteria of type two. It is worth further investigations on the effects of different refinement criteria. Additionally, a buffer area should be considered for new level to make sure that the flow structures requiring the high grid resolution won t move out of the boundary of fine level before next adjustment of the AMR grid during the computation. Not all elements belonging to the set Ω n can be flagged as the candidates for new nesting level, the set of elements to be flagged for constructing the blocks on new level k + 1, denoted by Ω r, is finally determined by Ω r = Ω a Ω n. (4.7) Step three: clustering and generating the new blocks After determining the elements to be refined, we should separate these elements to a number of distinct clusters to assure the required hierarchy. A parameter used to control the generation of the clusters is specified by the usage ratio of certain cluster as Usage ratio = number of flagged elements total number of elements. (4.8) At first, all elements belonging to the set Ω r are contained by a single cluster. The parameter usage ratio is then checked and compared with the threshold given by user in advance. If the lower limiter isn t achieved, the cluster will be split into two clusters. A key point here is how to find the optimal splitting point

4.1. BERGER S AMR SCHEME 147 to generate new clusters efficiently. One simple method is bisecting the cluster in the direction with larger length. A more smart method is proposed in [1] based on the method used in computer vision and pattern recognition. The cluster which is under the consideration for splitting is taken as a binary image f i j with the value of 1 for flagged elements and the value of for non-flagged ones. The horizontal and vertical signatures are specified as H j = f i j and V i = f i j. (4.9) i j We then look for the zero crossings in the 2 nd -order derivatives of signatures, which are called infection points. If more than one infection points are found for some cluster, the infection point with largest difference is selected as splitting point. After obtaining two new clusters, we turn to check their usage ratios. This procedure will repeated until all clusters satisfy the preselected threshold or maximal number of available clusters has been reached, which is also a preselected parameter by user. A block on the new level k + 1 is constructed based on the each cluster directly if all elements within this cluster belong to the set of Ω a. Otherwise, the cluster will be split again until all elements involved belong to the Ω a. At last, the initial condition for new-generated blocks is decided and the code will turn to set up the blocks on nest fine level if necessary. Adjusting AMR grid during the computation AMR grid should be adjusted during computation to track the change of the physical field every several time steps. In present scheme, we re-generate the AMR grid after each base grid step when the variable time step is used for different nesting

148 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME levels. The re-generation procedure begins from the highest nesting level. The operation order is different from that adopted during the initial setup and helpful to obtain higher accuracy [13]. Most numerical operations are same to what have been used in initial setup. Here only the differences will be described. The word old is used here to denote the existing grid at preset time step and new for the grid after adjustment. 1. Considering the task to generate the new blocks on nesting level k, the first work is to find out the set of elements Ω c which is determined by the blocks on the coarser levels. During initial setup, it is directly found by using block information on level k 1 which has already existed before the setup of level k. Because the adjusting procedure is conducted in a reverse order, the new blocks on level k 1 don t exist when we are generating the nesting level k. The set Ω c can be only determined by the finest level which isn t going to be changed in present adjusting operation, denoted by lev b here. When shrinking the set Ω c to set Ω a, enough space should be reserved for blocks on levels from lev b + 1 to k 1 to satisfy the required nesting relation. As a result, the domain shrinking should be repeated for several times. 2. Only the elements covered by old blocks on level k 1 will be checked by refinement criterion to find out Ω n and the buffer area is also included as initial setup. An additional step should be performed in adjusting operation is the set Ω n will be properly enlarged by taking the nesting relation between level k and k + 1 into account, because new blocks on level k + 1 usually have already existed if level k isn t the finest nesting level. 3. After constructing the new AMR grid, the solution arrays are then neces-

4.1. BERGER S AMR SCHEME 149 sary to be determined for new-generated blocks. For the elements in newgenerated blocks which are also covered by old blocks on the same resolution level, the solution is directly copied from the old elements. For the elements which are just generated by refining the larger ones of old blocks on next coarse level, the interpolation operations between old coarse elements and fine new ones are carried out to determine the flow solution. 4.1.3 Updating procedure on the AMR grid Each block can be independently updated during the computation if the values of ghost cells are properly provided. To set up the ghost cells, solution values of ghost cells will be copied from the neighboring blocks if they are located on the same nesting level as the host block or interpolated from neighboring blocks with the next coarse resolution. Updating procedure with fixed time step Being a simple case, the same time integration interval t is adopted for all blocks, which should satisfy the CFL stability condition of the smallest grid spacing. At the time instant t n we set up the necessary ghost cells for each block, the solution on each block then can be put forward to time instant t n+1 = t n + t by using the chosen numerical schemes. Post-processing operations Since the fine blocks overlay the blocks on the next coarse level, the solution in overlapping area is calculated with different grid resolutions. To keep the accuracy of numerical scheme, the solution on coarse block obtained by using large grid spacing will be replaced by the result calculated with fine grid resolution. This replacement procedure begins from the finest nesting level, the solution of blocks on this level will be used to cover the solution of next

15 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME coarse level in overlapping area. The procedure is repeated until the solution of level 2 modifies the solution of base grid. Figure 4.4: Conservation correction along coarse-fine boundary. Another post-processing operation is conservation corrections between the coarse and fine levels. As shown in top panel of Fig.4.4, boundary between level k and k + 1 is considered as an example and the refinement ratio of 2 is chosen for sake of simplicity. On coarse level k, the numerical flux e k i+1 j are adopted to update the volume-integrated average V q k i j, whereas its neighboring elements on

4.1. BERGER S AMR SCHEME 151 level k + 1, i.e. V q k+1 2i+1 2 j 1 and V q k+1 2i+1 2 j, are put forward by using flux e k+1 2i+1 2 j 1 and e k+1 2i+1 2 j. In general, following relation doesn t hold e k i+1 j = 1 2 ( e k+1 2i+1 2 j 1 + ek+1 2i+1 2 j). (4.1) To preserve the conservation of numerical scheme, the VIA moment of coarse element should be modified by ( ) V q k mod i j = V q k i j + t [ hk e k i+1, j V k h ( k+1 e k+1 2i+1,2 j 1 + ek+1 2i+1,2 j)], (4.11) where V k = (h k ) 2 is the area of the computational element on nesting level k. For another case shown in bottom panel of Fig.4.4, flux flows from fine level to coarse level. The VIA moment for coarse element is corrected by ( ) V q k mod i j = V q k i j t [ hk e k i j V k h ( k+1 e k+1 2i 1,2 j 1 + ek+1 2i 1,2 j)]. (4.12) The conservation corrections in y-direction are carried out with the similar formulations. Variable time step It isn t an efficient way to update all blocks using same time interval. The coarser blocks usually allow the larger time step. A more complex updating procedure should be adopted when different time integration intervals are chosen for blocks on different nesting levels. A recursive procedure should be used to update the flow solution on AMR grid. The variable t lev is used to store the time instant corresponding to the latest solution on nesting level lev. After putting forward the solution of nesting level k with an interval of t k, we determine the next level to be updated l as if k isn t the existing finest level then l = k + 1

152 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME else l=k do while t k = t k 1 l = k 1 conservation corrections between level k 1 and k modify the solution on level k 1 in overlapping area if l = 1, finish this base grid step end do end if 4.2 Implementation of AMR for MM-FVM scheme As a good candidate for AMR applications, multi-moment scheme applies Berger s AMR scheme straightforwardly. An advantage of using multi-moment concept is the localized spatial reconstruction in multi-moment schemes makes it possible to utilize single-cell based stencil for interpolations between coarse and fine levels, which avoids the interpolation operations involving more than two nesting levels. In this section, we introduce the multi-moment interpolations between coarse and fine levels and present some numerical tests for two dimensional transport problems. The other aspects of constructing an AMR-version multi-moment scheme are almost same to what have been done for single-moment method described in previous section. 4.2.1 Coarse-fine interpolation based on multi moments The coarse-fine interpolation operations are necessary for supplementing the ghost cells in advancing procedure and determining the flow solutions of new-generated fine elements during adjusting the AMR grid. The two tasks can be illustrated by Fig.4.5 and Fig.4.6, respectively.

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 153 Figure 4.5: Coarse-fine interpolation for supplementing ghost cells. Fig.4.5 shows the supplementation of ghost points for fine blocks along the boundary between level k and k + 1, where the refinement ratio is chosen to be 2. The known PVs (including the auxiliary PV at the cell center) on coarse grid are denoted by hollow circles. The ghost PVs required to be evaluated are shown by small solid markers. Three kinds of markers classify the ghost PVs into three types. The ghost points denoted by solid circles are located at the points where the known PV moments on coarse block are defined, the values are directly copied from the known PVs. The ghost points denoted by solid squares are located on the grid lines of coarse block, the PVs are determined by using the one dimensional interpolation polynomials for corresponding line elements. The last type, ghost PVs denoted by solid triangles are internal points of the coarse element. Generally, two dimensional reconstruction polynomial is needed. But we can compute the PVs on these points by using the one dimensional interpolatants based on

154 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME ghost PVs of type two. This method is equivalent to implement a multidimensional reconstruction by using one dimensional polynomials to sweep the different directions one by one. To limit the stencil for interpolation within a single cell, we use MM-FVM 3 profile for the problems without considering monotone property. For problems where the monotone reconstruction is necessary to avoid the spurious oscillations, MM-FVM M profile should be adopted. Considering the scheme given in chapter 2 requires additional information from neighboring cells for computing the slope limiter, we use a more compact algorithm for slope limiter as d M i = minmod (d L, d R ), (4.13) where d L = 2 x ( V q i P q i 1 2 ) and dr = 2 x where the denotations used are same as shown in Fig.2.1. ( P q i+ 1 2 V q i ), (4.14) The task of determining the flow solution for new-generated fine blocks is shown in Fig.4.6, coarse control volume C k i j is covered by four fine elements C k+1 2i 1,2 j 1, Ck+1 2i 1,2 j, Ck+1 2i,2 j 1 and Ck+1 2i,2 j. Different from above case, the unknowns include both PV and VIA moments. The values of PV moments can be determined by using the same method as mentioned above. The method used for evaluating the VIA moments (denoted by solid diamonds) is described as follows. The single-cell based reconstruction profile based on two PV moments and one VIA moment is represented by RF ( P q l, P q r, V q ) for sake of simplicity. The line-integrated averages over the four boundary edges of the coarse element, i.e. L q l, L q rl q b, L q t for left, right, bottom and top edges respectively, are computed by

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 155 Figure 4.6: Coarse-fine interpolation for determining the flow solution of new generated fine elements. using three-point Simpson s rule based on known PV moments as L q l = 1 6 L q r = 1 6 L q b = 1 6 L q t = 1 6 ( ) P q k i 1 2 j 1 + 4P q k 2 i 1 2 j + P q k i 1 2 j+ 1 2 ) ( P q k i+ 1 2 j 1 2 + 4P q k i+ 1 2 j + P q k i+ 1 2 j+ 1 2 ( P q k i 1 2 j 1 2 + 4P q k i j 1 2 + P q k i+ 1 2 j 1 2 ( P q k i 1 2 j+ 1 2 + 4P q k i j+ 1 2 + P q k i+ 1 2 j+ 1 2 Then the following quantities are obtained V q l = 2 x xi x i 1 2 RF V q r = 2 xi+ 1 2 RF x x i, (4.15), (4.16) ), (4.17) ). (4.18) ( ) L q l, L q r, V q k i j dx, (4.19) ( ) L q l, L q r, V q k i j dx, (4.2) where V q l, V q r are volume-integrated averages over left and right halves of the

156 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME control volume C k i j and L q tl = 2 x xi x i 1 2 RF L q tr = 2 xi+ 1 2 RF x L q bl = 2 x x i xi x i 1 2 RF L q br = 2 xi+ 1 2 RF x x i (P q ki 12 j+ 12, P q ki+ 12 j+ 12, L q t ) dx, (4.21) (P q ki 12 j+ 12, P q ki+ 12 j+ 12, L q t ) dx, (4.22) (P q ki 12 j 12, P q ki+ 12 j 12, L q b ) dx, (4.23) (P q ki 12 j 12, P q ki+ 12 j 12, L q b ) dx, (4.24) where L q tl, L q tr are line-integrated averages over the left and right halves of top boundary edge and L q bl, L q br are corresponding quantities on bottom boundary edge. Based on the quantities given above, the VIA moments for new-generated fine elements are determined by one dimensional interpolatants as V q k+1 2i 1,2 j 1 = 2 y j RF ( ) L q y bl, L q tl, V q l dy, (4.25) y j 1 2 V q k+1 2i,2 j 1 = 2 y j RF ( ) L q y br, L q tr, V q r dy, (4.26) y j 1 2 V q k+1 2i 1,2 j = 2 y j+ 1 2 RF ( ) L q y bl, L q tl, V q l dy, (4.27) y j V q k+1 2i,2 j = 2 y y j+ 1 2 y j RF ( L q br, L q tr, V q r ) dy. (4.28) 4.2.2 Numerical tests for two dimensional transport Before testing the practical AMR tests, the characteristics of numerical errors of MM-FVM scheme on overset grid are checked by numerical examples [67]. The computational grid is configured as follows. Two nesting levels are constructed on the computational domain [.5,.5] [.5,.5]. The base grid has

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 157 the uniform resolution of 1 8 in both x and y directions. One block has the resolution of level 2, which is located at the center of the computational domain and the area is 1. The refinement ratio between level 1 and level 2 is k. The AMR grid 4 remains unchanged during the computations for this special test. Two dimensional advection problem (2.95) is solved on this grid by using MM-FVM 3 schemes. The initial condition is specified as q (x, y, ) = sin (2πmx) with u = 1, v =, (4.29) where m is wavenumber of initial condition and m = 1 is adopted in this test. Four tests are carried out with different computational parameters as follows. Case 1: refinement ratio k = 2; constant time interval for different levels of t = 1, i.e. CFL=.1 for level 1 and CFL=.2 for level 2 8 Case 2: refinement ratio k = 2; variable time interval for different levels, CFL=.2 for both levels Case 3: refinement ratio k = 3; constant time interval for different levels of t = 1, i.e. CFL= 1 for level 1 and CFL=.2 for level 2 12 15 Case 4: refinement ratio k = 3; variable time interval for different levels, CFL=.2 for both levels The numerical results using MM-FVM 3 scheme with RK-3 temporal integration are given in Fig.4.7 and Fig.4.8 for above four different tests. We can find from the results that increase of resolution decelerates the wave propagation and using variable time interval for different nesting levels doesn t distort the numerical results obviously.

158 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME.4.2 Y -.2 -.4 -.4 -.2.2.4 X.4.2 Y -.2 -.4 -.4 -.2.2.4 X.4.2 Y -.2 -.4 -.4 -.2.2.4 X Figure 4.7: Sine wave advection on overset grid with refinement ratio of 2, shown are initial condition (top) and numerical results at t=1 for test 1 (middle) and for test 2 (bottom).

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 159.4.2 Y -.2 -.4 -.4 -.2.2.4 X.4.2 Y -.2 -.4 -.4 -.2.2.4 X.4.2 Y -.2 -.4 -.4 -.2.2.4 X Figure 4.8: Sine wave advection on overset grid with refinement ratio of 3, shown are initial condition (top) and numerical results at t=1 for test 3 (middle) and for test 4 (bottom).

16 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME The results can be explained through analysis on phase speed of adopted scheme. By using Von Neumann method for one dimensional linear equation as shown in 2.1.3, the absolute phase speed errors at different wavenumbers are shown in Fig.4.9 for MM-FVM 3 schemes, where different curves are obtained by using different resolutions and CFL numbers corresponding to above tests. MM- FVM 3 scheme is accelerating, phase error decreases with the increasing resolution and is not sensitive to the change of CFL number. The normalized phase speeds of different cases at wavenumber of 1 are given in Table 4.1. The error of phase speed is well controlled to a low level by using CIP/multi-moment concept. The conclusion is same as what has been pointed out in [67]. Phase speed error.6.5.4.3.2 No.1: x=1/8, CFL=.2 No.2: x=1/8, CFL=.1 No.3: x=1/8, CFL=.67 No.4: x=1/16, CFL=.2 No.5: x=1/24, CFL=.2.1 5 1 15 2 25 3 35 4 Wavenumber Figure 4.9: Absolute errors of phase speed, shown are results of MM-FVM 3 scheme with different CFL numbers and grid resolutions. Table 4.1: Normalized phase speed of MM-FVM 3 scheme with different resolutions and CFL numbers. Case No. 1 2 3 4 5 Norm phase speed 1% 99.998% 99.998% 99.885% 99.878%

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 161 The AMR-version multi-moment scheme is then tested by simulating the two dimensional transport problems as follows. The gradient-based refinement criterion is adopted here. An element is necessary to be refined if max ( x i j, y i, j) > δ where δ is a preselected threshold, x i j and y i j are evaluated by x i j = P q i+ 1 2 j P q i 1 2 j y i j = P q i j+ 1 2 P q i j 1 2 (4.3) The fist example is the advection of a square wave, the initial field is { 1, if max ( x, y ) <.25 q (x, y, ) =, otherwise, (4.31) in the computational domain [ 1, 1] [ 1, 1], the flow field is specified by u = 1, v = 1. The numerical tests are carried out on seven different grids, i.e. 4 1 1, 4 2 2, 4 3 2, 4 2 4, 8 1 1, 8 2 2, 16 1 1. Here the grid is denoted by n maxlv k, where n is the resolution of the base grid (uniform grid spacing over the computational domain), maxlv is the maximal nesting level and k is refinement ratio (keeping constant between different levels). The gradientbased threshold of.2 is selected and the parameter usage ratio is set as 8% in this test. The normalized errors are checked for all cases. The tests running on the non- AMR grids, i.e. 4 1 1, 8 1 1 and 16 1 1, provide the reference solutions for verifying the AMR scheme. The normalized errors and CPU time consumptions of seven cases are listed in Table 4.2, 4.3 and 4.4, where the results are organized according to the finest resolution. Obviously the results reveal that the normalized errors are converged and computational costs are reduced as ex-

162 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME pected. Numerical results on grids 4 2 2 and 4 3 2 are illustrated in Fig.4.1 and Fig.4.11 by mesh and contour plots respectively. Table 4.2: Normalized errors and CPU time of the square wave advection on the grid with resolution of 4 4. Grid l 1 l 2 l CPU time 4 1 1.4584.375.6359 1 Table 4.3: Same as Table 4.2, but for the tests on grids with resolution of 8 8. Grid l 1 l 2 l CPU time 8 1 1.2794.294.6654 8.6 4 2 2.285.2892.664 3.5 Table 4.4: Same as Table 4.2, but for the tests on grids with resolution of 16 16. Grid l 1 l 2 l CPU time 16 1 1.1671.2239.6943 76.4 4 3 2.1678.222.6793 19.9 4 2 4.1675.2236.7146 17.8 8 2 2.1677.2227.6499 25.4 The square wave advection with constant flow field is a very special case for present AMR scheme due to the discontinuous edges are always parallel to the grid lines. Under such condition, it is easy to generate the clusters which are required to be rectangles in Berger s AMR scheme. Two more challenging test cases are also carried out for further verifying the AMR implementation. Additional two cases are specified as the advection of a cylinder { 1, if x2 + y q (x, y, ) = 2 <.25, otherwise, (4.32) with constant velocity of u = 1, v = 1 and the solid rotation (u = 2y, v = 2x) of

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 163 a square wave { 1, if max ( x.35, y ) <.25 q (x, y, ) =, otherwise. (4.33) According the normalized errors given in Table 4.5,4.6,4.7 for cylinder case and Table 4.8,4.9,4.1 for solid rotation case, the agreement between the errors on AMR grids and those on non-amr grids again validates the present scheme. The numerical results on two AMR grids of 4 2 2 and 4 3 2 for above two test cases are shown in Fig.4.12 and Fig.4.14 by mesh plots and in Fig.4.13 and Fig.4.15 by contour plots. Compared with the first test case, the structure of AMR grid is much more complicated. In other words, more blocks are needed to construct the AMR grids which fit the physical field very well. The parameter usage ratio plays an important role in controlling the complexity of the grid. A lower requirement for parameter usage ratio results in a simpler grid with less blocks, but more unnecessary elements are refined to keep the hierarchy of the grid. Contrarily, if a larger threshold for parameter usage ratio is adopted, the structure of grid gets more complex, but the total number of refined elements decreases. The proper value of the parameter usage ratio should be selected by mainly considering the computational efficiency. These two tests show that the Berger s AMR scheme is capable of treating the complex physical fields. Table 4.5: Normalized errors and CPU time of the cylinder advection on the grid with resolution of 4 4. Grid l 1 l 2 l CPU time 4 1 1.3984.3193.453 1

164 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME Table 4.6: Same as Table 4.5, but for the tests on grids with resolution of 8 8. Grid l 1 l 2 l CPU time 8 1 1.233.2419.4293 8.6 4 2 2.2318.241.4368 3.5 Table 4.7: Same as Table 4.5, but for the tests on grids with resolution of 16 16. Grid l 1 l 2 l CPU time 16 1 1.1377.1879.4319 76.2 4 3 2.1387.188.4338 2.4 4 2 4.1372.1879.4278 18. 8 2 2.1386.1883.439 26.4 Table 4.8: Normalized errors and CPU time of square wave solid-rotation on the grid with resolution of 4 4. Grid l 1 l 2 l CPU time 4 1 1.3994.3539.6819 1 Table 4.9: Same as Table 4.8, but for the tests on grids with resolution of 8 8. Grid l 1 l 2 l CPU time 8 1 1.241.2724.77 8.6 4 2 2.242.2721.7134 3.9 Table 4.1: Same as Table 4.8, but for the tests on grids with resolution of 16 16. Grid l 1 l 2 l CPU time 16 1 1.1415.27.76 78.1 4 3 2.1423.275.727 22.7 4 2 4.1421.277.727 2.3 8 2 2.1423.275.7254 28.4

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 165 Figure 4.1: Mesh plots of numerical results of square wave advection on AMR grid 4 2 2 (top) and 4 3 2 (bottom).

166 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X Figure 4.11: Contour plots of numerical results of square wave advection on AMR grid 4 2 2 (left) and 4 3 2 (right). Shown are numerical results at different time instants of t = (top), t = 1 (middle) and t = 2 (bottom).

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 167 Figure 4.12: Same as Fig.4.1, but for the advection of a cylinder.

168 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X Figure 4.13: Same as Fig.4.11, but for the advection of a cylinder.

4.2. IMPLEMENTATION OF AMR FOR MM-FVM SCHEME 169 Figure 4.14: Mesh plots of numerical results of square wave solid-rotation on AMR grid 4 2 2 (top) and 4 3 2 (bottom).

17 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X 1 1.5.5 Y Y -.5 -.5-1 -1 -.5.5 1 X -1-1 -.5.5 1 X Figure 4.15: Contour plots of numerical results of square wave solid-rotation on AMR grid 4 2 2 (left) and 4 3 2 (right). Shown are numerical results at different time instants of t = 1π (top), t = 3 π (middle) and t = π (bottom). 5 5

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 171 4.3 AMR MM-FVM SWE model on cubed sphere In this section, the AMR-version multi-moment finite volume model is extended to spherical geometry with the application of the cubed-sphere grid. The cubed sphere consists of six patches which are logically identical to six computational domains covered by Cartesian grid if being considered in computational space. The AMR model using Berger s implementation can be applied on cubed-sphere grid straightforwardly. The key point for this extension is how to transfer solution values and block information among six patches, in other words, six base grids or computational domains. 4.3.1 Extension of AMR model to spherical geometry To construct the AMR grid on every patch, the block information from its contiguous patches is necessary, for example in determining the element set Ω a as descried in section 4.1. During the updating procedure, the information transfer across patch boundaries is needed to calculate the values of ghost cells and correct PV moments along patch boundaries as described in chapter 3. These procedures become more complicated for AMR model because blocks of different grid resolutions are possible to share the patch boundaries. Additional work for conservation corrections in AMR scheme depends on transferring block information among different patches since the operations are possible to occur between the blocks on different patches. To accomplish all these tasks, a proper numbering of all patches, patch boundaries and vertices is important. One option adopted in our implementation is shown in Fig.4.16, where for sake of simplicity, the sphere is represented by its

172 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME inscribed cube. The numbering of the patch isn t shown, it is same as definition in Fig.3.1. Twelve patch boundaries are numbered as L 1 to L 12 and eight vertices are V 1 to V 8. The local coordinates (ξ, η) are also shown in this figure on six patches. And it is should be pointed out that the local numbering of patch boundaries on each patch is using number 1, 2, 3 and 4 to represent the left, top, right and bottom boundaries. Figure 4.16: The numbering of patches, vertices and patch boundaries on cubed sphere. All information about the numbering is stored in a table for computer code to look for the necessary information. The content of this table is shown in Table 4.11 and 4.12. The explanation is given as follows.

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 173 Table 4.11: Content of numbering information table (part one). Global No. Vertex 1 No. Vertex 2 No. Patch 1 No. local No. L 1 V 5 V 1 1 1 L 2 V 6 V 2 1 3 L 3 V 7 V 3 2 3 L 4 V 8 V 4 3 3 L 5 V 1 V 2 1 2 L 6 V 2 V 3 2 2 L 7 V 3 V 4 3 2 L 8 V 4 V 1 4 2 L 9 V 5 V 6 1 4 L 1 V 6 V 7 2 4 L 11 V 7 V 8 3 4 L 12 V 8 V 5 4 4 Table 4.12: Content of numbering information table (part two). Global No. Patch 2 No. local No. Tangent Dir. Normal Dir. L 1 4 3 1 1 L 2 2 1 1 1 L 3 3 1 1 1 L 4 4 1 1 1 L 5 5 4 1 1 L 6 5 3 1-1 L 7 5 2-1 -1 L 8 5 1-1 1 L 9 6 2 1 1 L 1 6 3-1 1 L 11 6 4-1 -1 L 12 6 1 1-1

174 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME Each patch boundary is shared by two patches. We denote the patch with smaller number as Patch 1, and another as Patch 2. The direction of patch boundary is decided by one of the Patch 1 s coordinate directions, which is parallel to this boundary. Two vertices are found on each boundary, the Vertex 1 is defined as boundary s start point and Vertex 2 is end point. The Tangent Dir. has the value of 1 when the coordinate directions along the patch boundary of two contiguous patches are same and value -1 for different directions. Similarly, the parameter Normal Dir. is designed for the relation of two coordinate directions normal to the boundary patches. After the information transfer over the patch boundaries is accomplished with the application of the numbering information, the computation is independently carried out on each patch.

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 175 4.3.2 Numerical tests Three numerical tests are carried out for validating the AMR multi-moment SWE model on cubed sphere. The AMR model runs in these tests on different AMR grids. As in previous section, the normalized errors are checked to see if the numerical results convergence as expected. Meanwhile, the CPU time consumptions for different calculations are recorded to tell if the computational efficiency has been improved after using AMR technique. Both transport and shallow water flows are tested on the cubed sphere using AMR-version multi-moment model. Cosine bell advection on cubed sphere The advection of the cosine bell [85] is repeated on the AMR grid. The computational parameters are defined same as those in chapter 3, but MM-FVM P scheme is used in simulations with the RK-4 temporal integration here. The mesh refinement is based on the gradient of height field. An element C i j is necessary to be refined if max ( h x i j, hy i j) > δ, (4.34) where h x i j = P h i+ 1 2 j P h i 1 2 j h y i j = P h i j+ 1 2 P h i j 1 2. (4.35) The cosine bell advection test runs on seven different grids with resolution of 16 1 1, 32 1 1, 64 1 1, 16 2 2, 16 3 2, 16 2 4 and 32 2 2. The denotation of grid using n maxlv k is same to previous section, except the base grid is n n 6 on cubed sphere. The advections in two directions of α = π 2 and α = π are considered here. 4

176 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME The normalized errors are given in Table 4.13, 4.14, 4.15 for flow with α = π 2 and Table 4.17, 4.18, 4.19 for α = π. The normalized errors of tests on AMR grids 4 convergence to those on non-amr grids. The corresponding CPU time consumptions on different grids for different flow directions are give in Table 4.16 and 4.2. The normalized errors and the CPU time consumptions verify the performance of AMR-version multi-moment finite volume model on the cubed-sphere grid. The flow in the direction of α = π 4 is also important on AMR grid. Because the cosine bell passes four vertices and two complete patch boundaries, which are difficult for generating the AMR grid. The numerical results of flow with α = π 4 on two grids of 16 2 2 and 16 3 2 are given in Fig.4.17 and 4.18 at three representative instants of day 3, day 7.5 and day 12 (one complete revolution). Table 4.13: Normalized errors of cosine bell advection in the direction of α = π 2 on grid with resolution of 16 16 6. No. Grid l 1 l 2 l 1 16 1 1.1212E+.925E-1.9193E-1 Table 4.14: Same as Table 4.13, but for the tests on grids with resolution of 32 32 6. No. Grid l 1 l 2 l 2 32 1 1.1766E-1.1497E-1.1488E-1 3 16 2 2.1766E-1.1496E-1.1488E-1 Table 4.15: Same as Table 4.13, but for the tests on grids with resolution of 64 64 6. No. Grid l 1 l 2 l 4 64 1 1.336E-2.34E-2.4939E-2 5 16 3 2.3371E-2.3394E-2.4888E-2 6 16 2 4.3367E-2.34E-2.4933E-2 7 32 2 2.3369E-2.3396E-2.495E-2

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 177 Table 4.16: CPU time for cosine bell advection in the direction of α = π on 2 different AMR grids. Case No. 1 2 3 4 5 6 7 CPU time 1 7.34 2.29 59.46 8.64 8.75 14.32 Table 4.17: Normalized errors of cosine bell advection in the direction of α = π 4 on grid with resolution of 16 16 6. No. Grid l 1 l 2 l 1 16 1 1.9625E-1.7913E-1.118E+ Table 4.18: Same as Table 4.17, but for the tests on grids with resolution of 32 32 6. No. Grid l 1 l 2 l 2 32 1 1.1497E-1.1251E-1.1425E-1 3 16 2 2.1497E-1.1251E-1.1425E-1 Table 4.19: Same as Table 4.17, but for the tests on grids with resolution of 64 64 6. No. Grid l 1 l 2 l 4 64 1 1.32E-2.381E-2.3719E-2 5 16 3 2.3211E-2.376E-2.3743E-2 6 16 2 4.327E-2.38E-2.3757E-2 7 32 2 2.321E-2.377E-2.3737E-2 Table 4.2: CPU time for cosine bell advection in the direction of α = π on 4 different AMR grids. Case No. 1 2 3 4 5 6 7 CPU time 1 7.29 2.33 6.19 8.65 8.31 14.34

178 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 Figure 4.17: Numerical results of cosine bell advection (α = π 4 ) on 16 2 2 grid at day 3 (top), day 7.5 (middle) and day 12 (bottom).

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 179 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 Figure 4.18: Same as Fig.4.17, but for the test on 16 3 2 grid.

18 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME Non-smooth deformational flow A non-smooth deformational flow test on sphere was proposed in [48]. The flow field is specified same to (3.113), but with parameters of ρ = 2γ cos θ 1 + sin θ and ω = 3 3sech 2 ρ tanh ρ. (4.36) 2 cos θ The advected field is given as ( ) ρ φ = tanh sin λ, (4.37) γδ and the analytic solution is ( ρ φ = tanh γδ sin ( λ ω t ) ). (4.38) The parameters of γ = 1.5 and δ =.2 are set to obtain a non-smooth initial. This test is chosen due to its complicated flow structures. The MM-FVM M scheme is adopted to solve this discontinuous problem. Same as previous one, the gradient of height field is used as refinement criterion and simulations run on seven different grids. The normalized errors and corresponding CPU time consumptions shown in Table 4.21, 4.22, 4.23 and 4.24 give the same conclusion as the cosine bell advection test on the performance of the AMR multi-moment model. The structures of height field and corresponding AMR grids at different time instants are shown in Fig.4.19 and Fig.4.2 for the simulations running on 16 2 2 and 16 3 2 grids. Considerably complicated flow structures are observed in this test. During the computations the AMR grids have been properly adjusted by tracking the areas with large gradient in computational domain. The Berger s AMR scheme is capable of dealing with this complicated flow structures.

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 181 Table 4.21: Normalized errors of non-smooth deformational flow on grid with resolution of 16 16 6. No. Grid l 1 l 2 l 1 16 1 1.5127E-1.1199E+.6144E+ Table 4.22: Same as Table 4.21, but for the tests on the grids with resolution of 32 32 6. No. Grid l 1 l 2 l 2 32 1 1.2813E-1.8731E-1.65E+ 3 16 2 2.2818E-1.8738E-1.65E+ Table 4.23: Same as Table 4.21, but for the tests on the grids with resolution of 64 64 6. No. Grid l 1 l 2 l 4 64 1 1.158E-1.664E-1.511E+ 5 16 3 2.1659E-1.697E-1.5173E+ 6 16 2 4.159E-1.666E-1.511E+ 7 32 2 2.1654E-1.696E-1.5167E+ Table 4.24: CPU time for non-smooth deformational flow on different AMR grids. Case No. 1 2 3 4 5 6 7 CPU time 1 7.36 5.77 59.57 27.84 31.5 3.83

182 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18 Figure 4.19: Numerical results of non-smooth deformational flow on 16 2 2 grid at t =.625 (top), t = 1.5625 (middle) and t = 2.5 (bottom).

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 183 9 6 3-3 -6-9 -18-135 -9-45 45 9 135 18-135 -9-45 45 9 135 18-135 -9-45 45 9 135 18 9 6 3-3 -6-9 -18 9 6 3-3 -6-9 -18 Figure 4.2: Same as Fig.4.19, but for the test on 16 3 2 grid.

184 CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME Barotropic instability test The AMR-version global shallow water model has also developed based on the framework described in chapter 3. The barotropic instability test [22] is repeated by AMR model since the physical field changes with large gradient in a limited area near the patch boundaries between patch five and its neighbors. The criterion for refinement based on relative vorticity is adopted in this test. An element is refined if ζ = 1 [ 1 ( ) P 1 ( u Gi j η i+ 1 2 j P u i 1 2 j P v ξ i j+ 1 ) ] P v 2 i j 1 2 > δ, (4.39) where δ is a preselected threshold. MM-FVM 4 scheme is used to solve this test. The normalized l 2 errors on different grids are given in Fig.4.21 where the errors are computed by comparing the numerical results with reference one obtained by present model on 256 1 1 grid. According to CPU time consumptions on different grids shown in Table 4.25, the scheme keeps the accuracy of non-amr model while the computational efficiency is considerably improved by using AMR technique. The numerical results on three different grids 32 2 2, 32 3 2 and 128 1 1 are illustrated by contour plots in Fig.4.22 for height field at day 4, Fig.4.23 for day5 and Fig.4.24 for day 6. The meshes are too fine to be properly displayed, so only the boundaries between different nesting levels are shown in these figures.

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 185 1 1 1 NORM l 2 ERROR 1 1 1 2 1 3 1 4 1 5 1 2 3 4 5 6 DAY No.1 32 1 1 No.2 64 1 1 No.4 128 1 1 No.3 32 2 2 No.5 32 3 2 No.7 64 2 2 No.6 32 2 4 Figure 4.21: Time history of normalized l 2 errors of barotropic instability test on different grids. Table 4.25: CPU time for barotropic instability test on different AMR grids. Case No. 1 2 3 4 5 6 7 CPU time 1 7.23 2.91 57.15 17.55 14.95 2.58

CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 186 9 7 5 3 1-1 -18-135 -9-45 45 9 135 18-18 -135-9 -45 45 9 135 18-18 -135-9 -45 45 9 135 18 9 7 5 3 1-1 9 7 5 3 1-1 Figure 4.22: Numerical results of barotropic instability test at day 4. Shown are obtained on three different grids of 32 2 2 (top), 32 3 2 (middle) and 128 1 1 (bottom).

4.3. AMR MM-FVM SWE MODEL ON CUBED SPHERE 187 9 7 5 3 1-1 -18-135 -9-45 45 9 135 18-18 -135-9 -45 45 9 135 18-18 -135-9 -45 45 9 135 18 9 7 5 3 1-1 9 7 5 3 1-1 Figure 4.23: Same as Fig.4.22, but for the results at day 5.

CHAPTER 4. AMR MULTI-MOMENT FVM SCHEME 188 9 7 5 3 1-1 -18-135 -9-45 45 9 135 18-18 -135-9 -45 45 9 135 18-18 -135-9 -45 45 9 135 18 9 7 5 3 1-1 9 7 5 3 1-1 Figure 4.24: Same as Fig.4.22, but for the results at day 6.