5,988 research outputs found
High-order numerical solutions using cubic splines
The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes
Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations
We introduce a WENO reconstruction based on Hermite interpolation both for
semi-Lagrangian and finite difference methods. This WENO reconstruction
technique allows to control spurious oscillations. We develop third and fifth
order methods and apply them to non-conservative semi-Lagrangian schemes and
conservative finite difference methods. Our numerical results will be compared
to the usual semi-Lagrangian method with cubic spline reconstruction and the
classical fifth order WENO finite difference scheme. These reconstructions are
observed to be less dissipative than the usual weighted essentially non-
oscillatory procedure. We apply these methods to transport equations in the
context of plasma physics and the numerical simulation of turbulence phenomena
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