A Smoothness Indicator for Adaptive Algorithms for Hyperbolic Systems

@article{Karni2002ASI,
  title={A Smoothness Indicator for Adaptive Algorithms for Hyperbolic Systems},
  author={Smadar Karni and Alexander Kurganov and Guergana Petrova},
  journal={Journal of Computational Physics},
  year={2002},
  volume={178},
  pages={323-341}
}
The formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed. In this paper, a new smoothness indicator (SI) is used to identify “rough” solution regions and is implemented in locally adaptive algorithms. The SI is based on the weak local truncation error of the approximate solution. It was recently reported in S. Karni and A. Kurganov, Local error… 

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References

SHOWING 1-10 OF 27 REFERENCES

Primitive, Conservative and Adaptive Schemes for Hyperbolic Conservation Laws

For the last two or three decades, it has become an accepted practice to utilise conservative methods when solving numerically hyperbolic conservation laws. Shock waves are the solution features that

A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

A high-order extension of the recently proposed second-order, semidiscrete method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations

New Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations are introduced, based on the use of more precise information about the local speeds of propagation, and are called central-upwind schemes.

Third order nonoscillatory central scheme for hyperbolic conservation laws

Summary. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory

Non-oscillatory central differencing for hyperbolic conservation laws

Adaptive mesh refinement for hyperbolic partial differential equations

This work presents an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques based upon Richardson-type estimates of the truncation error, which is a mesh refinement algorithm in time and space.

Pointwise Error Estimates for Scalar Conservation Laws with Piecewise Smooth Solutions

We introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation (and, in fact, more general approximations) to scalar conservation laws with piecewise smooth

Local error estimates for discontinuous solutions of nonlinear hyperbolic equations

Let $u(x,t)$ be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose $u_\varepsilon (x,t)$ is the solution of an approximate viscosity

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

It is proved that a scalar version of the high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions.