Modified third and fifth order WENO schemes for inviscid compressible flows
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Modified third and fifth order WENO schemes for inviscid compressible flows Naga Raju Gande1
· Ashlesha A. Bhise1
Received: 27 February 2020 / Accepted: 26 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The weighted essentially non-oscillatory schemes are well known for their shock capturing abilities due to their properties resulting from weighted combination reconstruction taken such that less weight is given to less smooth stencils. In this article, contrary to this property, the analysis has been done by adding extra weight to less smooth substencils of the domain that in turn adds the useful information which plays a vital role in improving resolution of the solution mainly at discontinuities or sharp gradients. The theoretical aspects have been supported with scalar, one-dimensional as well as two-dimensional test problems for third- and fifth-order schemes. Numerical solutions obtained by the proposed third-order scheme are comparable to the numerical solutions obtained using some of the native fifth-order WENO schemes. Keywords WENO scheme · Nonlinear weights · Critical points · Approximation order · Euler equations · Two-dimensional Riemann problems
1 Introduction The study of hyperbolic conservation laws was originated as a wider class of Euler equations. At present it has become one of the most interesting research topic because of its modelling physical phenomena. The generic form of conservation laws is Ut + F(U)X = 0, X ∈ Rn , t > 0; U0 (X) =: U(X, 0), X ∈ Rn .
(1.1)
where U is the conserved physical vector quantity and F is the flux vector and n is the dimension of the space. The conserved quantity can neither be created nor be destroyed; however, in a phase space, it can be changed due to flow across boundary. Hence, the equation may permit discontinuous solutions. Sometimes it becomes impossible to calculate the exact solution of the problem containing discontinuities. Naga Raju Gande
[email protected] 1
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, India
Numerical Algorithms
In this case, numerical techniques are one of the powerful ways of finding the solution approximately up to the required order of accuracy. For decades, this has been one of the most challenging problems as a consequence of the occurrence of shocks, rarefaction waves, contact discontinuities etc. It is because the discontinuities in the solution often gave birth to undesirable oscillations or smear the sharpness of the discontinuities in the numerical solution. To address them, authors in [1] came up with the weighted essentially non-oscillatory (WENO) scheme that was based on the semi-discrete approximation (method of lines). There the spatial domain was divided into stencils of length, say, 2r − 1, which were further divided into substencils of length r. The reconstruction of the flux function was then done on the r point substencils. A weighted convex combination of the reconstructed numerical flux of all substencils was inclu
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