6 research outputs found
A modified fifth-order WENO scheme for hyperbolic conservation laws
This paper deals with a new fifth-order weighted essentially non-oscillatory
(WENO) scheme improving the WENO-NS and WENO-P methods which are introduced in
Ha et al. J. Comput. Phys. (2013) and Kim et al., J. Sci. Comput. (2016)
respectively. These two schemes provide the fifth-order accuracy at the
critical points where the first derivatives vanish but the second derivatives
are non-zero. In this paper, we have presented a scheme by defining a new
global-smoothness indicator which shows an improved behavior over the solution
to the WENO-NS and WENO-P schemes and the proposed scheme attains optimal
approximation order, even at the critical points where the first and second
derivatives vanish but the third derivatives are non-zero.Comment: 23 pages, 14 figure
Arc length based WENO scheme for Hamilton-Jacobi Equations
In this article, novel smoothness indicators are presented for calculating
the nonlinear weights of weighted essentially non-oscillatory scheme to
approximate the viscosity numerical solutions of Hamilton-Jacobi equations.
These novel smoothness indicators are constructed from the derivatives of
reconstructed polynomials over each sub-stencil. The constructed smoothness
indicators measure the arc-length of the reconstructed polynomials so that the
new nonlinear weights could get less absolute truncation error and gives a
high-resolution numerical solution. Extensive numerical tests are conducted and
presented to show the performance capability and the numerical accuracy of the
proposed scheme with the comparison to the classical WENO scheme.Comment: 14 pages, 9 figure
Numerical schemes for a class of nonlocal conservation laws: a general approach
In this work we present a rather general approach to approximate the
solutions of nonlocal conservation laws. Thereby, we approximate in a first
step the nonlocal term with an appropriate quadrature rule applied to the
spatial discretization. Then, we apply a numerical flux function on the reduced
problem. We present explicit conditions which such a numerical flux function
needs to fulfill. These conditions guarantee the convergence to the weak
entropy solution of the considered model class. Numerical examples validate our
theoretical findings and demonstrate that the approach can be applied to
further nonlocal problems
Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws
The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC
scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not
achieve the desired order of convergence in smooth monotone regions and at
critical points. In this article, this drawback is achieved with the WENO-LOC
smoothness indicators by constructing a WENO-Z type nonlinear weights which
contains a novel global smoothness indicator. This novel smoothness indicator
measures the derivatives of the reconstructed flux in a global stencil, as a
result, the proposed numerical scheme could decrease the dissipation near the
discontinuous regions. The theoretical and numerical experiments to achieve the
required order of convergence in smooth monotone regions, at critical points,
the essentially non-oscillatory (ENO), the analysis of parameters involved in
the nonlinear weights like and are studied. From this study, we
conclude that the imposition of certain conditions on and , the
proposed scheme achieves the global order of accuracy in the presence of an
arbitrary number of critical points. Numerical tests for scalar, one and
two-dimensional system of Euler equations are presented to show the effective
performance of the proposed numerical scheme.Comment: 25 pages, 10 figure