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Generalized Plane Waves (GPWs) were introduced to take advantage of Trefftzmethods for problems modeled by variable coefficient equations. Despite thefact that GPWs do not satisfy the Trefftz property, i.e. they are not exactsolutions to the governing equation, they instead satisfy a quasi-Trefftzproperty: they are only approximate solutions. They lead to high ordernumerical methods, and this quasi-Trefftz property is critical for theirnumerical analysis.  The present work introduces a new family of GPWs, amplitude-based. Themotivation lies in the poor behavior of the phase-based GPW approximations inthe pre-asymptotic regime, which will be tamed by avoiding high degreepolynomials within an exponential. The new ansatz is introduces higher orderterms in the amplitude rather than the phase of a plane wave as was initiallyproposed. The new functions' construction and the study of their interpolationproperties are guided by the roadmap proposed in [16]. For the sake of clarity,the first focus is on the two-dimensional Helmholtz equation withspatially-varying wavenumber. The extension to a range of operators allowingfor anisotropy in the first and second order terms follows. Numericalsimulations illustrate the theoretical study of the new quasi-Trefftzfunctions.

Amplitude-based Generalized Plane Waves: new Quasi-Trefftz functions for scalar equations in 2D