1. Stratified Magnetohydrodynamics Accelerated Using GPUs:SMAUG

2. The Sheffield Advanced Code The Sheffield Advanced Code (SAC) is a novel fully non-linear MHD code based on the Versatile Advection Code (VAC)  designed for simulations of linear and non-linear wave propagation  with gravitationally strongly stratified magnetised plasma.  Shelyag, S.; Fedun, V.; Erdélyi, R. Astronomy and Astrophysics, Volume 486, Issue 2, 2008, pp.655- 662 Shelyag, S.Fedun, V.Erdélyi, R.

3. Full Perturbed MHD Equations for Stratified media

4. Numerical Diffusion Central differencing can generate numerical instabilities Difficult to find solutions for shocked systems We define a hyperviscosity parameter which is the ratio of the forward difference of a parameter to third order and first order Tracking evolution of the hyperviscosity we can identify numerical noise and apply smoothing where necessary

5. Why MHD Using GPU’s? F(i-1,j+1)F(I,j+1)F(i+1,j+1) F(i-1,j)F(i,j)F(i+1,j) F(i-1,j-1)F(i,j-1)F(i+1,j-1) Excellent scaling with GPU’s but, Central differencing requires numerical stabilisation Stabilisation with GPU’s trickier, requires Reduction/maximum routine An additional and larger mesh Consider a simplified 2d problem Solving flux equation Derivative using central diffrencing Time step using Runge Kutta

6. Halo Messaging Each proc has a “ghost” layer – Used in calculation of update – Obtained from neighbouring left and right processors – Pass top and bottom layers to neighbouring processors Become neighbours ghost layers Distribute rows over processors N/nproc rows per proc – Every processor stores all N columns SMAUG-MPI implements messaging using a 2D halo model for 2D and 3D halo model for 3D Consider a 2d model – for simplicity distribute layers over a line of processes

7. Processor 1 Processor 2 Processor 3 Processor 4 N+1 1 N p2 min p3 max p2 min p1 min p2 max p1 min Send top layer Send bottom layer Receive top layer Receive bottom layer

8. MPI Implementation Based on halo messaging technique employed in SAC code void exchange_halo(vector v) { gather halo data from v into gpu_buffer1 cudaMemcpy(host_buffer1, gpu_buffer1,...); MPI_Isend(host_buffer1,...,destination,...); MPI_Irecv(host_buffer2,...,source,...); MPI_Waitall(...); cudaMemcpy(gpu_buffer2,host_buffer2,...); scatter halo data from gpu_buffer2 to halo regions in v }

9. Halo Messaging with GPU Direct void exchange_halo(vector v) { gather halo data from v into gpu_buffer1 MPI_Isend(gpu_buffer1,...,destination...); MPI_IRecv(gpu_buffer2,...,source...) MPI_Waitall(...); scatter halo data from gpu_buffer2 to halo regions in v } Simpler faster call structure

10. Progress with MPI Implementation Successfully running two dimensional models under GPU direct – Wilkes GPU cluster at The University of Cambridge – N8 - GPU Facility, Iceberg 2D MPI version is verified Currently optimising communications performance under GPU direct 3D MPI implementation is already implemented still requires testing

11. Orszag-Tang Test 200x200 Model at t=0.1, t=0.26, t=0.42 and t=0.58s

12. A Model of Wave Propagation in the Magnetised Solar Atmmosphere The model features a Flux Tube with Torsional Driver, with a fully stratified quiet solar atmosphere based on VALIIIC Grid size is 128x128x128, representing a box in the solar atmosphere of dimensions 1.5x2x2Mm Flux tube has a magnetic field strength of 1000G Driver Amplitude 200km/s

13. Timing for Orszag-Tang Using SAC/SMAUG with Different Architetures

14. Performance Results (Hyperdiffusion disabled) Grid Size (number of GPUs in brackets) With GPU direct ( time in s) Without GPU direct (time in s) 1000x1000(1)31.5431.5 1000x1000(2x2)11.2811.19 1000x1000(4x4)12.8913.7 2044x2044(2x2)41.341.32 2044x2044(4x4)42.443.97 4000x4000(4x4)77.3777.44 8000x8000(8x8)63.361.7 8000x8000(10x10)41.941.0 Timings in seconds for 100 iterations (Orszag-Tang test)

15. Performance Results (With Hyperdiffusion enabled) Grid Size (number of GPUs in brackets) Without GPU direct (time in s) 2044x2044(2x2)184.1 2044x2044(4x4)199.89 4000x4000(4x4)360.71 8000x8000(8x8)253.8 8000x8000(10x10)163.6 Timings in seconds for 100 iterations (Orszag-Tang test)

16. Conclusions We have demonstrated that we can successfully compute large problems by distributing across multiple GPUs For 2D problems the performance using messaging with and without GPUdirect is similar. – This is expected to change when 3D models are tested It is likely that much of the communications overhead arises from routines used transfer data within the GPU memory – Performance enhancements possible through application architecture modification Further work needed with larger models for comparisons with X86 implementation using MPI The algorithm has been implemented in 3D testing of 3D models will be undertaken over the forthcoming weeks