An orthonormal basis set is introduced for use in generating global solutions to nonlinear differential equations with shocks. By global, it is meant that a single series solution is generated that is valid over the entire domain, not a separate solution across discretized elements of the domain. Its derivation was motivated by shortcomings in all current computational fluid dynamic algorithms for dealing with complex, hypersonic flows involving interacting shocks. The fractal-like derivation (infinitely self-similar) focused on the distribution of segment lengths yields a scaled set of Walsh functions.
Computational fluid dynamic simulation of hypersonic flows generally requires specialized algorithm modifications to accommodate shocks. Two fundamentally different approaches, shock fitting and shock capturing, have evolved to simulate shocks. The first approach, shock fitting, formally recognizes a shock as a discontinuity in the field. The orientation of the shock and its velocity in the domain are tracked as supplementary dependent variables in the solution. A shock may be tracked as a moving boundary as in the case of blunt body flow, or treated as a discontinuity that moves within the interior of a domain.
The second approach, shock capturing, formulates the governing flow equations in strong conservation form. Shocks are assumed to be computable to the extent that the representation of conserved flux is smoothly varying across discontinuities. In practice, flux reconstruction algorithms usually require some capability to detect captured shocks or slip surfaces, and employ appropriate limiters to maintain stability and suppress Gibbs phenomena. Shock capturing algorithms will generally require greater grid resources to achieve equivalent accuracy of a shock fitting algorithm assuming smooth regions of the flow are computed similarly.
An orthonormal basis set composed of Walsh functions is used for deriving global solutions (valid over the entire domain) to nonlinear differential equations that include discontinuities. It was thought that a basis function set composed of simple, discontinuous square waves may enable a more robust algorithm for detecting a shock normal direction even in the context of a finite-volume algorithm. As the orthonormal basis set evolved, it became clear that its self-mapping property under multiplication provided new capabilities for understanding nonlinear problems far beyond its original intent for multi-dimensional reconstruction. The global solution approach developed here solves for variables in wave number space, not physical space. The physical domain is not explicitly discretized, although an implicit discretization is engaged as a function of wave number. This approach provides new opportunities and challenges for capturing shocks while retaining their discontinuous structure.
The use of Walsh functions for solving nonlinear differential equations has been studied in the past. The current derivations offer a different perspective, focused on the relation of segment sizes in the basis functions and associated resolution requirements in computational fluid dynamic applications. The ability to accurately locate a discontinuity in a solution or to resolve the structure of a viscous shock wave is a primary focus of the present work.