Temporally Uniform Grid Convergence of Discrete Approximations and Numerical Simulations in the Problems of Wave Propagation over Unbounded Domains

North Carolina State University, Raleigh NC

Investigators

Abstract

Two major well-recognized difficulties in numerical simulation of waves propagating over unbounded domains are the accumulation of error during long time intervals and necessity to truncate the domain and subsequently set the artificial boundary conditions (ABCs) at the external artificial boundary. These two issues turn out to be closely related. In the previous work of the PI with collaborators that has been done in the framework of the scalar wave equation, we have used the inherently three-dimensional phenomenon of lacunae and developed a methodology that modifies any appropriate discrete scheme so that the long-term error buildup is fully eliminated while all of the original properties of the scheme (e.g., order of accuracy) are preserved. Moreover, the procedure allows one to replace the original infinite domain by a finite computational domain, which leads to obtaining highly accurate non-local unsteady ABCs. These ABCs are built directly for the discrete formulation of the problem, their temporal non- locality is fixed and limited, and they possess full geometric universality. The key objective of the proposed project is to extend the aforementioned methodology to wave-type models of practical interest, in particular, the Maxwell's equations (electromagnetic waves) and the linearized Euler's and full-potential equations (acoustic waves). The attainability of this goal is accounted for by the fact that the solutions to these equations have sharp aft fronts of the waves (manifestation of lacunae), which is the exact same behavior as displayed by the solutions to the wave equation. Numerical simulation of waves on unbounded domains has numerous applications that range from scattering of electromagnetic waves from aircraft and ground vehicles (radar technology) to antenna design to calculation of the acoustic fields produced by the airframe and airplane's jet engines for the purpose of reducing the noise levels around airports, as well as inside the passenger compartments. The results that we expect to obtain are going to benefit the foregoing applied areas. Indeed, we anticipate the creation of a universal framework that would allow one to modify a wide class of already existing and proven methods so that to enable two additional crucial capabilities -- long- term integration and accurate computation of infinite-domain wave fields on truncated domains. Besides, as the proposed research unfolds, it will necessarily include communication and collaboration with physicists and engineers, as well as preparation of course materials and training young researchers.