A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions.
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The analysis exposes that the effects of staircasing as well as a lack of properly enforced jump-conditions on the field components have significant consequences for the global accuracy.
The Riemann solution accounts for the relevant physics by enforcing appropriate jump conditions at the phase boundary.
The method furthermore avoids the use of any mixed states in discretization stencils, by reconstructing the pure fluid states in each mixed cell, explicitly enforcing the jump conditions across the interface.
The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary.
Moreover, the evolution equations ensure automatically the proper contact conditions for chemical potentials at the migrating interface enforcing the jumps in chemical potentials being the same for all components.
Fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point.
