Efficient Numerical Algorithms for Heterogeneous Computation of PDE Extended Systems with Applications

Partial Differential Equations (PDEs) and corresponding numerical schemes are explored to simulate scientific and engineering problems including parachute simulation and American option pricing. These problems involve appropriate coupling of several equations systems. A revised spring-mass model is used to describe the motion of parachute canopy and string motion which considers both string stiffness and angular stiffness. This model is validated by the material's Young's modulus and Poisson ratio and is proved to be convergent to continuum mechanics. The Navier-Stokes equation is applied to simulate the fluid field and a second-order accurate numerical scheme is used, together with the introduction of the concept "penetration ratio" to simulate fabric porosity which has great impact on the drag performance of the parachute. A partial-integro differential equation based on generalized hyperbolic distribution is built to simulate the price of American option pricing after coupling certain free boundary condition to describe early exercise property.;Due to the complex nature of above applications and the corresponding numerical scheme structure, Graphics Processing Unit (GPU) is introduced to derive efficient heterogeneous computing algorithms. The most computationally intensive and parallelizable parts of the application are identified and accelerated greatly based on the single-instruction multiple data (SIMD) architecture. During the parallelization process, parallel execution, memory hierarchy and instruction usage are optimized to maximize parallelization effect. For the spring-mass system, we achieved 6 times speedup and greatly improved the parachute simulation efficiency. The system of one-dimensional gas dynamics equations is solved by the Weighted Essentially Non-Oscillatory (WENO) scheme; the heterogeneous algorithm is 7-20 times faster than the pure CPU based algorithm. For single American option, the numerical integrations are parallelized at grid level and 2 times speedup is realized; for multiple option pricing, each thread is in charge of one option and the algorithm reaches 400 times speedup.