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Parallel Computation of Shallow Water Flows Using Hybrid MPI/OpenACC

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29 January 2024

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Abstract
A parallel shallow water flow model is introduced in this paper. The explicit-time finite volume approach is adopted to solve the 2D shallow water equations on an unstructured triangular mesh. The proposed scheme is second-order accurate in temporal and spatial terms using the two-stage Runge-Kutta and the monotone upwind scheme for conservation law (MUSCL) methods, respectively. Based on Message Passing Interface (MPI) and OpenACC, a multi-GPU model is presented with the METIS library to produce the domain decomposition. A CUDA-aware MPI library through GPUDirect for peer-to-peer (P2P) transfer between two GPUs and overlapping computation and MPI communication are used to speed up MPI memory exchange and the performance of the code. A 2D circular dam break test with wet and dry downstream beds and grid resolutions of about 2 million cells is considered to verify the accuracy of the code, and good results were achieved compared to the numerical simulations of published studies. Compared with the multi-CPU version of the 6-core CPU, maximum speedups of 56.18 and 331.51 were obtained using the single-GPU and multi-GPU versions, respectively. Results indicate that acceleration performance improves as the mesh resolution increases.
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Subject: Environmental and Earth Sciences  -   Water Science and Technology

Introduction

Numerous numerical models have been developed during the last two decades to simulate floods using the finite volume method to solve shallow water equations (SWEs) in two dimensions (2D) (Bradford and Sanders 2002; Begnudelli and Sanders 2007; Song, Zhou, Guo, et al. 2011; Hou et al. 2015). Shallow water modeling is computationally expensive, especially for computing large domains and mesh refinement to improve the numerical scheme's stability and accuracy (Lai and Khan 2017; Hu and Song 2018). Parallel computing using GPUs can be implemented to overcome this problem. Often, GPU codes can achieve a performance that cannot be obtained by standard CPU systems (Caplan et al. 2019). Generally, GPU programming models are CUDA (NVIDIA 2022), OpenCL (Group 2022), and OpenACC (OpenACC 2022). Since CUDA and OpenCL are low-level programming approaches, users are required to develop the code each time according to the GPU specifications (Xue and Roy 2021). Rewriting code may be quite challenging for developers in terms of programming time and knowledge of computer architecture systems (Caplan et al. 2019). OpenACC is a high-level programming approach, that works only by adding directives to the code. The compiler, such as the NVIDIA HPC SDK, will automatically thread the parallel regions to the GPUs (Norman 2022).
OpenACC has been successfully applied to many 2D hydrodynamic models. Herdman et al. (2012) CloverLeaf mini application accelerated with the OpenACC programming model, achieving a 4.91 speedup with the NVIDIA X2090 GPU over a 16-core CPU. Zhang et al. (2016) the flow field case was performed using an OpenACC parallel computing application, with a speedup of 18.6 achieved by employing the NVIDIA Quadro K2000 GPU compared to a 4-core CPU. Zhang et al. (2017) proposed a 2D parallel dam break model using OpenACC applications and obtained 20.7 speedups using the NVIDIA Tesla K20c GPU versus a 4-core CPU.
To obtain better computation performance, multiple GPUs may be needed. In this work, a hybrid MPI/OpenACC framework is developed with domain decomposition. CPUs are defined as hosts that manage control instructions and file I/O. GPUs are defined as devices that handle the most computationally intensive parts of the code.
In this paper, a parallel approach using hybrid MPI and OpenACC on multiple GPUs is developed to tackle the 2D shallow water problem using an explicit-time finite volume model on an unstructured triangular mesh. A domain decomposition is presented employing METIS (Karypis and Kumar 1997) and CUDA-aware MPI (Kraus 2013). In addition to second-order spatial accuracy, the scheme also implements second-order temporal accuracy to avoid extremely diffusive results. The parallel performance of the developed in-house code (Saleem and Mohammad 2020) examined by benchmark test cases.
The article is structured as follows: In Section 2, given the governing equations for SWEs. The SWEs are then discretized using an explicit-time Godunov-type finite volume technique on a triangular grid, as described in Section 3. In Sections 4 and 5, domain decomposition and parallel implementation are provided, respectively.

Shallow water equations

The 2D SWEs in its conservative version proposed by Song, Zhou, Guo, et al. (2011); Song, Zhou, Li, et al. (2011)
∂ U ∂ t + ∂ E ∂ x + ∂ G ∂ y = S ,
The variable vectors U , E , G , and   S are given by
U = h h u h v , E = h u h u 2 + 1 2 g h 2 h u v , G = h v h u v h v 2 + 1 2 g h 2 ,
S = S 0 + S f = 0 g h S 0 x g h S 0 y − 0 g h S f x g h S f y
in which h indicates the water height;   u and   v are velocities in the x and y directions, respectively; the gravitational acceleration is defined by g ; ( S 0 x = − ∂ z / ∂ x , S 0 y = − ∂ z / ∂ y ) and ( S f x , S f y ) denote the bed and friction sources in the x and y directions, respectively; here z represents the bed elevation.
S f x = n 2 u u 2 + v 2 h 4 / 3 , S f y = n 2 v u 2 + v 2 h 4 / 3
where n is the coefficient of roughness.

Numerical scheme

Finite volume discretization on unstructured triangular grids for 2D SWEs
Equation (1) can be written in integral form as
∂ ∂ t ∫ Ω   U d Ω + ∫ Ω   ∂ E ∂ x + ∂ G ∂ y d Ω = ∫ Ω   S d Ω
Equation (4) can be rewritten by applying the divergence theorem as
∂ ∂ t ∫ Ω   U d Ω + ∮ Γ   F · n d Γ = ∫ Ω   S d Ω
where Ω , Γ , and n denote the domain, boundary, and outward normal vector as defined by n x , n y T , respectively. F · n represent the flux vector normal to the boundary and is given by
F U · n = h u ⊥ h u u ⊥ + 1 2 g h 2 n x h v u ⊥ + 1 2 g h 2 n y
herein u ⊥ = u n x + v n y is the velocity perpendicular to the cell face. The line integral of the F U · n over all faces of the triangular cell may be represented algebraically as
∮ Γ   F U · n d Γ = ∑ k = 1 3 F k U L , U R · n k L k
where k   and L are the index and length of the face, respectively. U L , U R indicates Riemann states. The discretization of equation (5) in an arbitrary triangular cell C i is
Ω i d U i d t = − ∑ k = 1 3 F i , k U L , U R · n i , k L i , k + S i
in which Ω i and U i denote the area and average variables of the cell C i , respectively; i   is the index of the cell.

3.1. Time integration

The two-stage explicit Runge-Kutta scheme (Liang and Marche 2009; Liang 2010; Song, Zhou, Guo, et al. 2011; Song, Zhou, Li, et al. 2011; Hou et al. 2013; Hou et al. 2015) is employed to achieve second-order temporal accuracy, and the new time step of equation (8) is obtained by
U i n + 1 = 1 2 U i n + U i n * + L i U n *
where
U i n * = U i n + L i U n
And L i is given by
L i U = − ∆ t Ω i ∑ k = 1 3 F i , k · n i , k L i , k − S i , o + S i , f

3.2. Stability criteria

The stability of the explicit scheme on a triangular grid is handled by selecting an appropriate time step using the Courant-Friedrichs-Lewy (CFL) proposed by Delis, Nikolos, and Kazolea (2011); Delis and Nikolos (2013). If R i is the minimum distance from the ith triangle’s centroid to its faces, the CFL condition is expressed as
∆ t = C F L   m i n R i u i 2 + v i 2 + g h i
In this paper, the numerical scheme is based on an explicit-time, finite volume Godunov-type cell-centered approach on an unstructured triangular mesh. The Harten-Lax-Van-Leer-Contact (HLLC) approximate Riemann solver (Toro 2001; Liang and Borthwick 2009; Song, Zhou, Guo, et al. 2011; Hou et al. 2013) is utilized to calculate the cell interface fluxes. The current numerical method similar to Song, Zhou, Guo, et al. (2011) incorporates the sloping bottom model as well as volume-free surface relationships (VFRs) (Begnudelli and Sanders 2006) to give an efficient and robust scheme for tracking stationary or moving wet/dry fronts. A water depth reconstruction technique (Begnudelli and Sanders 2007; Song, Zhou, Guo, et al. 2011) is adopted, to tackle the trapping of small water volumes on sloping beds. Second-order accuracy in spatial is obtained by employing the MUSCL method (Van Leer 1979; Hubbard 1999). The adaptive method proposed by Hou et al. (2013) is applied to relieve unphysically high velocities produced by the MUSCL reconstruction near the wet/dry interfaces. Source terms are separated into bed slope and friction components, as expressed in equation (2). The bed slope terms are precisely discretized depending on the water level at the cell centroid, and the well-balanced feature is maintained, as explained by Song, Zhou, Guo, et al. (2011). The sources of friction are resolved via a semi-implicit method (Song, Zhou, Guo, et al. 2011; Bi et al. 2015).

Domain decomposition and halo cells

The preprocessing step generates the 2D unstructured triangular mesh for the whole domain, similar to Saleem and Mohammad (2020). The whole domain mesh is decomposed into subdomains by applying the mesh’s graph’s partitioning tools from the METIS library (Karypis and Kumar 1997). The goal is to minimize the contact surfaces between subdomains to reduce processor communication costs (Lai and Khan 2017). To perform data exchange between subdomains using the MPI library, each subdomain is typically given one layer of halo cells or ghost cells. The halo cells in this paper are generated as follows: (i) For each subdomain, the boundary edges are identified. (ii) The nodes of boundary edges are compared with the nodes of other subdomains’ boundary edges, and a list of shared nodes are created. (iii) The cells of each subdomain are investigated against the shared nodes list; the cells that have two consecutive nodes in the shared list are the sent cells for the compared subdomain and the received cells, or halo cells, for the neighbor subdomain.
After finding send and receive cells for each subdomain, the subdomain cells are renumbered as follows: the interior cells will be sent or received first, followed by the cells that will be sent to adjacent subdomains, and the halo cells or received cells from the neighboring subdomains at the end.

Parallel implementation

Simulating shallow water flows using a finite volume scheme is computationally intensive. Since the explicit time-stepping procedure is adopted, and within a single time step the numerical mesh calculation is uncorrelated. Therefore, the loop computation on each grid can be performed in parallel (Zhang et al. 2017). In this study, the hybrid MPI/OpenACC is applied to leverage multiple GPUs on a single node. MPI is a parallel programming approach (Forum 2021) that allows data to be shared between processors through messages (Xue, Jackson, and Roy 2021). OpenACC is a high-level programming model (OpenACC 2022) that utilizes directives and clauses to automatically parallelize loops (Hu and Song 2018). The OpenACC parallel loop directive with the level of parallelism gang, worker, vector, and seq clauses are adopted in the code to offload loops to the GPU automatically (OpenACC 2022). For tightly nested loops, OpenACC loop collapsing is employed to combine many loops into one loop, which significantly improves on-node parallelization efficiency (Zhang et al. 2019). The OpenACC routine directive is utilized to call functions or subroutines inside the parallel loops (OpenACC 2022). The async directive, which decreases kernel launch overhead, is another key OpenACC directive used in the code (Wright et al. 2021), and the OpenACC wait statement is adopted to control OpenACC queuing (Norman 2022). The reduction clause is used to manage race conditions in the code when multiple parallel threads access the same memory location in a way that may lead to incorrect results (Norman 2022).
As shown in Figure 1, boilerplate code (Colgrove 2021) is implemented to assign each MPI rank to a specified GPU. The OpenACC set device_num () function is utilized to set the desired GPU. Once the GPU number is set, all upcoming OpenACC directives will be sent to the selected GPU until another GPU is specified with the same function (OpenACC 2022). In this study, non-blocking communication is adopted using MPI_Isend and MPI_Irecv to exchange halo cells between subdomains, and blocking MPI_Waitall is called to check the completeness of the communication. The CUDA-aware MPI (Kraus 2013) library with GPUDirect for peer-to-peer (P2P) transfer is used to directly buffers between the memories of two GPUs on the same node. In OpenACC, the host_data use_device () is utilized before MPI calls to avoid CPU and GPU data transfer. To improve the performance, computation and MPI communication are overlapped. As proposed in Kraus (2022), for each subdomain, the computation of cells close to the boundary is performed, then the halo cells are exchanged between the subdomains concurrently while computing the internal cells.

Performance analysis

The developed numerical scheme is validated using 2D circular dam break tests. The simulations were conducted using Saturn Cloud, a data science platform managed by NVIDIA. It contains up to 8 NVIDIA Tesla V100 GPUs with Intel Xeon E5-2686 v4 processors running at 2.3 GHz (base) and 2.7 GHz (turbo); each GPU has 5120 cores and 16 GB of memory; connected through NVLink for peer-to-peer GPU communication (AWS 2022). The NVIDIA HPC SDK compiler was used to run the computations with double precision (64-bit).

1.1. Circular dam break

This hypothetical test was proposed by Alcrudo and Garcia-Navarro (1993). In the center of the 200 m by 200 m domain with a frictionless flatbed is a 50 m radius cylindrical dam. The initial condition of the water depth within the dam is 10 m, while it is set at 5, 1, and 0 m outside the dam. The dam wall is considered to be instantly removed to simulate a 2D circular dam break wave and the present model's shock-capturing capabilities. Figure 2 (a and b) show a coarse mesh and domain decomposition for the 2D circular dam break, respectively. The domain configurations that have been examined are 515564, 1031224, and 2062372 triangular cells. For all simulations, a 2-second time and CFL of 0.7 are adopted. The calculated 3D view and water depth contour following the circular dam break are shown in Figure 3 and Figure 4, with downstream depths of 5 and 1 m, respectively. Subcritical flow is present over the entire domain, at a downstream water level of 5 m. At a downstream water level of 1 m, there is a change from subcritical flow to supercritical flow, which is an interesting test for the developed scheme. The presented numerical scheme is efficient when dealing with the dry bed problem as shown in Figure 5. In all simulations, the circular waves spread out from the deepest part of the water in a radial and symmetrical way.
The results of total execution times for the CPU and GPU computations were reported using three different mesh resolutions as shown in Table 1. A single GPU's performance was optimized with a V100 GPU. To compare the acceleration of a single GPU to the entire CPU, a multi-CPU model with six MPI tasks employing each task per subdomain was developed using an Intel Core i7-8750H CPU (6 cores). Utilizing the same simulation time and test case, a 56.18, 37.44, and 24.31 times speedup with a single GPU against the multi-CPU version was obtained on the 2062372, 1031224, and 515564 grids, respectively. Also, the performance of a multi-GPU model was investigated by dividing the three grids over 4 and 8 GPUs, using one MPI task per GPU. A maximum speedup ratio between multi-GPU and multi-CPU versions is 331.51 and 182.33 with 8 and 4 GPUs, respectively, for 2062372 grid resolution. These results indicate that acceleration performance improves as the number of grid cells increases.
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Figure 2. The domain of the 2D circular dam break.
Figure 2. The domain of the 2D circular dam break.
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Figure 3. A circular dam breaks after 2 s with a 5m downstream water depth.
Figure 3. A circular dam breaks after 2 s with a 5m downstream water depth.
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Figure 4. A circular dam breaks after 2 s with a 1m downstream water depth.
Figure 4. A circular dam breaks after 2 s with a 1m downstream water depth.
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Figure 5. A circular dam breaks after 2 s with a dry downstream bed.
Figure 5. A circular dam breaks after 2 s with a dry downstream bed.
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Table 1. Runtimes and CPU/GPU speedup ratio.
Table 1. Runtimes and CPU/GPU speedup ratio.
Grid CPU
(6 cores)		 1 GPU 4 GPUs 8 GPUs
Runtime (s) Runtime (s) Speedup Runtime (s) Speedup Runtime (s) Speedup
515564 4668 192 24.31 75 61.24 53 88.08
1031224 25871 691 37.44 228 113.47 145 178.42
2062372 40113 714 56.18 220 182.33 121 331.51

Conclusion

A shallow water flow model using MPI and OpenACC has been developed to solve the 2D shallow water equations using the Godunov-type finite volume approach. The scheme is second-order accurate in both space and time.
The unstructured 2D triangular grid was used to implement the multi-GPU explicit-time-step finite volume model. Domain decomposition was performed by the METIS library, and to minimize the MPI message latency time, the sent and received cell numbering was especially considered for each subdomain.
Finally, the single-GPU and multi-GPU models' performances were compared to the multi-CPU model. The maximum speedup ratios obtained with 1 GPU, 4 GPUs, and 8 GPUs are 56.18, 182.33, and 331.51, respectively, versus a 6-core CPU with the largest grid resolution. It was concluded that as the resolution of mesh cells increases, so does the speedup ratio.

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Figure 1. Boilerplate code to select the GPU for each MPI rank (Colgrove 2021).
Figure 1. Boilerplate code to select the GPU for each MPI rank (Colgrove 2021).
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