High-Resolution Numerical Schemes For Conservation Laws With Discontinuous Flux

The equation of conservation laws with discontinuous flux in space,which can model traffic flow and two-phase flow,is broadly applied in multifarious fields.This kind of equation of conservation laws features discontinuous flux in some spatial place,which results in jump state of unknown function thereby.This fact brings about challenges of theoretical research on this kind of equation of conservation laws and designing numerical schemes for it,such as classical schemes with high-order accuracy for equation of conservation laws with continuous flux can’t straight expand to it.Although some classical first-order schemes have been modified and applied to this equation,they can’t exhibit effects of high-order accuracy and high resolution,especially near discontinuities in space.It is studied in this dissertation that numerical schemes with high-order accuracy and high resolution for this kind of equation of conservation laws.First of all,it is built that a new way of mesh dissection,which selects stationary discontinuity as grid interface instead of center;because of reconstruction with respect to unknown function may cause numerical oscillation near stationary discontinuity,flux reconstruction is considered.That means,restrictive continuity condition for numerical flux is imposed at stationary discontinuity,thus a kind of interfacial numerical flux which meets continuity condition is introduced and reconstructed by upwind-type fifth-order WENO method.This method can not only ensure high-order accuracy of numerical solution but also suppress numerical solution’s pseudo oscillation.To better maintain steady-state of numerical solution,a kind of equilibria-preserving limiter is utilized to modify reconstructed numerical flux.Then for improving extent of legibility and sharpness of numerical solutions near discontinuities,upwind-type fifth-order multi-resolution WENO flux reconstruction method is introduced.This kind of reconstruction method involves three nested stencils and weakens influence of those outer stencils that span across discontinuities by automatically minishing corresponding weights,and amplifies weight of reconstructed values upon inner stencils which is continuous for reconstructed objects.The advantage of this method is keeping high-order accuracy and achieving high resolution near discontinuities.Furthermore a fourth-order linear central compact scheme is applied to strengthen sharpness of transition of numerical solutions near discontinuities.Another idea is utilizing fifth-order multi-resolution WENO method to reconstruct unknown function,this reconstruction method can more effectively suppress pseudo oscillations of numerical solutions near discontinuities than ordinary fifth-order unknown function WENO reconstruction method.At last,The above-mentioned methods can be extended to one-dimensional system of conservation laws modeling polymer flooding of two-phase flow.Considering “the idea of discontinuous flux”,this kind of equation system is decoupled,then those high-order and high-resolution schemes proposed for scalar equation of conservation laws with discontinuous flux in space in former paragraphs is utilized to approach the first equation;And by viewing the second equation as linear advection equation,hyperbolic-tangenttype THINC function is introduced to interpolate and approach it.Corresponding numerical solutions demonstrate satisfying effects.The above schemes all temporally advance along a way of the third-order TVD-type Runge-Kutta discretization to strengthen suppression of pseudo oscillations.Some relative examples’ numerical results verify these newly constructed schemes’ effectiveness.Figure [40] table [15] reference [54]...