Modified cubic B-spline quasi-interpolation numerical scheme for hyperbolic conservation laws

ABSTRACT

In general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified version of the cubic B-spline quasi-interpolation (CBSQI) numerical scheme for the hyperbolic conservation laws in one space dimension. In order to stabilize the CBSQI scheme, we have added an adaptive artificial viscosity term and derived the conditions under which the modified scheme is monotone. Although the CBSQI scheme can be of at most fourth order accurate in space, the monotone condition restricts the resulting scheme to be of at most order of one, as expected. In order to improve the order of accuracy, we adopt a procedure, which we call the weighted adaptive viscosity approach, through which we achieved the fourth order accuracy both in space and time in the modified CBSQI scheme. Further, linear $L^2$-stability of the modified CBSQI scheme is analyzed using von-Neumann analysis. Numerical experiments are performed to validate the efficiency and stability of the modified CBSQI scheme.

COMMUNICATED BY:

• Constantin Bacuta

KEYWORDS:

• Artificial viscosity
• shock capturing method
• B-spline quasi-interpolation

AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 35D40

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Correction

Acknowledgements

The author also would like to thank the reviewers and Prof. S. Baskar for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Rakesh Kumar  http://orcid.org/0000-0002-5829-0384

Additional information

Funding

The author acknowledges the financial support of Council of Scientific and Industrial Research (CSIR), Government of India [grant number 09/087(0592)/2010-EMR-I].