Focus on solving wave equation: Make F ourier guess to get Helmholtz equation:

1. Focus on solving wave equation: • Make Fourier guess to get Helmholtz equation: • Solve Helmholtz equation (waves at single frequency) • Superimpose these solutions with time dependent coefficients to find solutions to the wave equation Traditionally, hyperbolic equations are solved numerically with grid-based methods, e.g. finite volume, differences methods However, these exhibit limitations [1]: • Becomes computationally heavy if k2has large range • Domain discretization is time consuming • Field solution must be produced Random Walk Method (RWM) is an attractive alternative that can overcome these difficulties Tara Aida1,2, Dr. Sorin Mitran1, Michael Malahe1 Introduction Investigate possibility of combing RWM with grid-based methods to solve wave equation and the Helmholtz equation. Compare the computational load of RWM with traditional methods for these solutions. Apply RWM to a specific problem in acoustics. Objectives Conclusion References Acknowledgements [1] M.K. Chati et al. “Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz’s equations.” 2001. Mathematics Department at UNC-Chapel Hill1 and Harvard University2 Random Walk Method Methods Finite Element Method (FEM) Gain familiarity with traditional techniques Solve Helmholtz equation with FEM, varying domain and k2 value Finer meshes lead to lower error, but require more computational power Conduct convergence study: analyze the effect of k2 value on the relation between error and mesh fineness CUDA Program for RWM Write parallelized code to solve the wave equation and the Helmholtz equation in inhomogeneous media Use GO language and CUDA platform on NVIDIA GPUs Conduct convergence study to compare with FEM study Apply RWM to Acoustics Apply RWM to solve hyperbolic PDEs in specific acoustics problem. Compare RWM to previous numerical solutions. Theory: Applies to second-order partial differential equations [1]. To find the solution v(x) at a point x in the domain: • Conduct random walks, starting at x, ending at the boundary • Calculate path integrals along walks • Solve for v(x) using formula1 relating it to boundary conditions, and expectation value of path integrals. Benefits of RWM: Allows one to solve PDE at specific points, rather than over entire domain Easily parallelized with low communication between each walk  lower computational load Through RWM, we hope to find solutions to the wave equation, within some error, and with relatively low computational power, even over a wide range of wave frequencies Since hyperbolic equations apply to many areas in physics, a faster more flexible method of solving these equations could benefit a wide range of computational research. This research made possible by NSF Award OCI-1156614. I’d also like to thank Dr. Mitranand Michael Malahe for their guidance as my main mentors, as well as Dr. Kannappan for her work in coordinating the CAP REU. Results Figure 3. Plot of relative error versus measure of mesh fineness, , for FEM solutions of the Helmholtz equation, with linear fits for k2 varying from 1 to 105, for domain shown. GPU Random Walk Method for Hyperbolic Problems Figure 2. Sample FEM solution for Helmholtz equation on custom domain, with . Figure 1. Two random walks produced using Mathematica random number generators. 1Derived from theory of diffusion processes, stochastic calculus, Itô formula