Linear finite-element numerical techniques for combustion problems requiring variable step size

Combustion problems frequently pose situations in which the dependent variables (temperature, species concentrations, etc.) vary rapidly in some local domain and much more slowly throughout the remaining region of interest. Such situations arise for both fluid-mechanical (e.g., boundary-layer behavior) and chemical reasons (i.e., Arrhenius- type temperature dependence of chemical reaction rates). In any case, they present a particular difficulty in the numerical solution of the governing conservation equations: if a fixed space-step is to be used, its magnitude is dictated by stability and/or accuracy requirements of the limited but rapidly varying domain. Outside this narrow domain, many more space steps are taken than are required; computing costs are thus magnified. Various means to avoid this difficulty have been used in the past, but none has proved generally satisfactory or convenient. Finite-element methods are substantially more convenient than the classical finite-difference techniques for approximating spatial dependencies in certain combustion problems. We discuss only linear finite-element (LFE) methods here. A discussion of this and more general finite-element methods can be found in several sources [1-3]; we briefly outline the application of the LFE method and illustrate its advantages

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events