International audienceThe non-steady heat equation is considered in thin structures. The asymptotic expansion of the solution constructed earlier is used for evaluation of the partial derivatives of the solution. The method of partial asymptotic domain decomposition is applied to the non-steady heat equation. It reduces the original 2D model to a hybrid dimension one, partially 2D, partially 1D with some special interface conditions between the 2D and 1D parts. The finite volume method is applied for the numerical solution of the hybrid dimension model. The error estimate is proved. The numerical experiment confirms the theoretical error evaluation
The nonoscillatory central difference scheme of Nessyahu and Tadmor is a Godunovtype scheme for one-dimensional hyperbolic conservation laws in which the resolution of Riemann problems at the cell interfaces is bypassed thanks to the use of the staggered Lax-Friedrichs scheme. Piecewise linear MUSCL-type (monotonic upstream-centered scheme for conservation laws) cell interpolants and slope limiters lead to an oscillation-free second-order resolution. Convergence to the entropic solution was proved in the scalar case. After extending the scheme to a two-step finite volume method for two-dimensional hyperbolic conservation laws on unstructured grids, we present here a proof of convergence to a weak solution in the case of the linear scalar hyperbolic equation ut + div(V u) = 0. Since the scheme is Riemann solver-free, it provides a truly multidimensional approach to the numerical approximation of compressible flows, with a firm mathematical basis. Numerical experiments show the feasibility and high accuracy of the method.
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