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A single nonlinear equation must be solved in each of four common arrangements of the clothoid.
The formulation requires that only a subset of the discrete fluid equations be solved in each iteration.
The governing equation is first discretized in time, leading to a time-stepping scheme, where a nonlinear exterior elliptic problem has to be solved in each time step.
The resulting set of discrete semi-smooth equations can be solved efficiently by a variant of Newton's method, where all additionally introduced variables are condensed from the global system so that a linear system only consisting of the displacement degrees of freedom has to be solved in each iteration step.
Using the well-known Lyapunov drift-plus-penalty policy, we extract the instantaneous problem that needs to be solved in each slot to control the data admission and allocate the time slots, power, and subchannels.
The conservation law that must be solved in each direction and at each time step can be formally written as frac{partial U}{partial x^{0"}} + frac{partial F}{partial x^{1"}} =0.
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A local boundary value problem (at a time level) is solved in each subdomain, with boundary values extracted from the neighboring domains.
More specifically, large systems of non-linear equations are solved in each time step by Newton iterations, typically contributing more than 80% of the total compute time.
To handle the dynamic nature of the problem, this study adopts a rolling horizon approach to break down the proposed problem into a set of stages, in which a static bike repositioning sub-problem is solved in each stage.
Species conservation differential equations as well as reaction and heat release rates provided by the S5 and S7 schemes are solved in each computational cell via a direct integration approach.
The mass range of the solid soot phase is divided into thirty-five disectionsections and two equations are solved in each section to model the formation of the fractal-like soot aggregates.
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