Entropy-Controlled Artificial Anisotropic Diffusion for the Numerical Solution of Conservation Laws Based on Algorithms from Image Processing |
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| Affiliation: | 1. School of Mathematics and Statistics, Xi''an Jiaotong University, Xi''an 710049, China;2. Department of Mathematics, Shaanxi University of Science and Technology, Xi''an 710021, China;3. Department of Mathematics, Comilla University, Cumilla 3506, Bangladesh;1. Department of Endocrinology and Metabolism, Institute of Endocrinology, Liaoning Provincial Key Laboratory of Endocrine Diseases, The First Affiliated Hospital of China Medical University, Shenyang 110001, Liaoning, China;2. Department of Endocrinology and Metabolism, The Sixth Affiliated People''s Hospital of Shanghai Jiaotong University, Shanghai 200233, China;3. Department of Endocrinology and Metabolism, China-Japan Friendship Hospital, Beijing 100029, China;4. Department of Endocrinology and Metabolism, Chinese People''s Liberation Army General Hospital, Beijing 100853, China;5. Department of Endocrinology and Metabolism, Sun Yat-sen University Third Hospital, Guangzhou 510630, Guangdong, China;6. Department of Endocrinology and Metabolism, Peking University People''s Hospital, Beijing 100044, China;7. Department of Endocrinology and Metabolism, Shanxi Provincial People''s Hospital, Taiyuan 030012, Shanxi, China;8. Department of Endocrinology and Metabolism, Sichuan University West China Hospital, Chengdu 610041, Sichuan, China;9. Department of Endocrinology and Metabolism, Fourth Military Medical University Xijing Hospital, Xi''an 710032, Shaanxi, China;10. Department of Endocrinology and Metabolism, Affiliated Drum Tower Hospital of Nanjing University Medical School, Nanjing 210008, Jiangsu, China;11. Department of Endocrinology and Metabolism, Qilu Hospital of Shandong University, Jinan 250012, Shandong, China;12. Department of Endocrinology and Metabolism, Peking University First Hospital, Beijing 100034, China;13. Department of Endocrinology and Metabolism, Henan Provincial People''s Hospital, Zhengzhou 450003, Henan, China;14. Department of Endocrinology and Metabolism, Second Affiliated Hospital of Harbin Medical University, Harbin 150001, Heilongjiang, China;15. Department of Endocrinology and Metabolism, Xiangya Second Hospital of Central South University, Changsha 410008, Hunan, China;p. Department of Endocrinology and Metabolism, Xinjiang Uygur Autonomous Region Hospital, Urumqi 830001, Xinjiang, China;q. Department of Epidemiology and Statistics, Institute of Basic Medical Sciences, Chinese Academy of Medical Sciences, School of Basic Medicine, Peking Union Medical College, Beijing 100730, China;1. Infectious Disease Responses Laboratory, University of New South Wales, Sydney, Australia;2. School of Mathematical and Physical Sciences, University of Technology Sydney, Australia |
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| Abstract: | In image processing are nonlinear anisotropic diffusion filters used to construct suitable filter algorithms for denoising, edge enhancement, and edge detection. We applied a nonlinear anisotropic diffusion operator in the context of the numerical solution of a scalar hyperbolic conservation law. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular we stabilize a central scheme, known for its oscillatory behavior, by the construction of a nonlinear diffusion term. This means constructing a diffusion matrix consisting of eigenvectors parallel and perpendicular to discontinuities and eigenvalues denoting the amount of dissipation depending on the local strength of the gradients. These directions are used to steer the amount of dissipation, which means suppressing diffusion across the shock front and using the perpendicular direction to enable the necessary diffusion to stabilize the underlying second-order scheme. This new approach allows a multidimensional view to the concept of artificial dissipation. We take the concept of entropy production in the vicinity of shock regions as an indicator for the steering of the diffusion matrix. In smooth regions, which obey an entropy equality instead of an entropy inequality, no information about the diffusion direction and strength is needed. So we add as a stabilizing term diffusion parallel to the characteristics, which turns out to be the Lax–Wendroff diffusion rate. In the case of unsteady regions we use the entropy production to blend between this diffusion term and an—still second-order—extra diffusion term with diffusion parallel to the gradient of the entropy production. |
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