Klein-Gordon equation with advection on unbounded domains using spectral elements and high-order non-reflecting boundary conditions

A reduced shallow water model under constant, non-zero advection in the infinite channel is considered. High-order (Givoli–Neta) non-reflecting boundary conditions are introduced in various configurations to create a finite computational space and solved using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems. The numerical solution of a wave propagation problem in a very large or unbounded domain provides a challenging computational difficulty – namely, solving the problem on a finite computational domain while maintaining the true essence of the solution. One of the modern techniques that has garnered a significant amount of attention in handling this challenge is the absorbing or non-reflecting boundary condition (NRBC) method. In using this method, the original infinite domain is truncated by an artificial boundary B, resulting in a finite computational domain X and the residual domain D. When truncating the domain, the modeler must devise boundary conditions for the truncated domain. Of course, by imposing a boundary where one does not physically exist, the problem is changed – and unless chosen carefully, would certainly be expected to pollute the solution as the problem evolves and impinges on the boundary. Therefore, two main possibilities exist for the modeler: Choose a convenient, easily implementable boundary condition that does not necessarily reflect the physical problem and solve it on a large sub-domain. The idea behind this technique is that the boundary effects are negligible for a short time evolution of the problem in a small area of interest away from the boundaries. Choose a boundary condition that preserves the true behavior of the infinite solution at the boundary and solve the problem on a smaller sub-domain. The idea behind this technique is that the additional effort extended to impose a better boundary condition will be worth the effort and allow for solving the problem on a smaller domain. For obvious reasons, the first possibility has only limited usefulness. To see why, suppose that we wanted to model the wave motion following a pebble dropped in the center of a large, still pond. Now, suppose that we have a truncated domain to model this phenomena – say a bathtub. If the pebble is dropped in the bathtub, the waves generated by the pebble would …