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ABSTRACT
A meshless local Petrov–Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov–Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 106 to 108. It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability.
Nomenclature
| D | = | node distance |
| x, y | = | special coordinates |
| n | = | unit normal vector outward to the boundary |
| = | heat source | |
| q | = | heat flux on the boundary |
| N(x) | = | shape function |
| N | = | total number of nodes |
| = | fictitious nodal values | |
| τ,τ1 | = | stability coefficient |
| I | = | unit operator |
| Tcom | = | computational values |
| Ladv | = | differential operator |
| Hs | = | Sobolev space |
| K | = | stiff matrix |
| w | = | test function |
| = | boundary | |
| Ω | = | problem domain |
| Pe | = | Peclet number |
| N | = | total number of nodes |
| T | = | temperature |
| Th(xI) | = | trial function |
| uj | = | velocity |
| λ | = | thermal conductivity |
| Ps | = | orthogonal projection operator |
| Tref | = | referential values |
| = | new test function | |
| k | = | Gauss integration nodes |
| F | = | coefficient vector |
Nomenclature
| D | = | node distance |
| x, y | = | special coordinates |
| n | = | unit normal vector outward to the boundary |
| = | heat source | |
| q | = | heat flux on the boundary |
| N(x) | = | shape function |
| N | = | total number of nodes |
| = | fictitious nodal values | |
| τ,τ1 | = | stability coefficient |
| I | = | unit operator |
| Tcom | = | computational values |
| Ladv | = | differential operator |
| Hs | = | Sobolev space |
| K | = | stiff matrix |
| w | = | test function |
| = | boundary | |
| Ω | = | problem domain |
| Pe | = | Peclet number |
| N | = | total number of nodes |
| T | = | temperature |
| Th(xI) | = | trial function |
| uj | = | velocity |
| λ | = | thermal conductivity |
| Ps | = | orthogonal projection operator |
| Tref | = | referential values |
| = | new test function | |
| k | = | Gauss integration nodes |
| F | = | coefficient vector |