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- Efraimsson, Gunilla, et al.
(author)
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A note on the effect of artificial viscosity on solutions of conservation laws
- 1996
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In: Applied Numerical Mathematics. - : Elsevier. - 0168-9274 .- 1873-5460. ; 21, s. 155-173
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Journal article (peer-reviewed)abstract
- We consider central difference schemes with artificial viscosity terms for nonlinear hyperbolic systems. We analyze the influence of artificial viscosity on solutions with shocks of nonlinear hyperbolic problems in one dimension. Both stationary and moving shocks are considered. The analysis shows that for the Euler equations one can obtain well behaved (sharp) shock layers when a scalar viscosity coefficient is used. This is not true for a general system. Numerical computations of Burgers' equation and the Euler equations are presented. They support the results from the linear theory.
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- Engquist, Björn, et al.
(author)
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Far field boundary conditions for computation over long time
- 1988
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In: Applied Numerical Mathematics. - 0168-9274 .- 1873-5460. ; 4:1, s. 21-45
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Journal article (peer-reviewed)abstract
- A new class of computational far field boundary conditions for hyperbolic partial differential equations is developed. These boundary conditions combine properties of absorbing boundary conditions for transient solutions and properties of far field boundary conditions for steady-state problems. The conditions can be used to limit the computational domain when both traveling waves and evanescent waves are present. Boundary conditions for scalar wave equations are derived and analyzed. Extensions to systems of equations are discussed and results from numerical experiments are presented.
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8. |
- Harten, A, et al.
(author)
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Some results on uniformly high-order accurate essentially nonoscillatory schemes
- 1986
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In: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274 .- 1873-5460. ; 2:3-5, s. 347-377
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Journal article (peer-reviewed)abstract
- We continue the construction and the analysis of essentially nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy in the sense of truncation error, at extrema of the solution. In this paper we construct an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory. This means that, for piecewise smooth solutions, the variation of the numerical approximation is bounded by that of the true solution up to O(hR - 1), for 0 <R and h sufficiently small. The design involves an essentially nonoscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. To solve this reconstruction problem we use a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities.
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9. |
- Lenferink, Wilhelmus
(author)
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Pointwise convergence of approximations to a convection-diffusion equation on a Shishkin mesh
- 2000
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In: Applied Numerical Mathematics. - 0168-9274 .- 1873-5460. ; 32:1, s. 69-86
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Journal article (peer-reviewed)abstract
- A centered difference or finite element discretization is applied to a singularly perturbed, one-dimensional boundary value problem. The discretization uses a piecewise equidistant mesh. It is proved that the pointwise error is (almost) of second order with respect to the number of nodes, uniformly in the perturbation parameter. The proof is based on a monotonicity argument.
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