Pub Date : 2023-01-01DOI: 10.4208/aamm.oa-2021-0019
Xiang Li, D. Ma, Nan-Sheng Liu null, Pei Wang
{"title":"A Compact Eulerian Interface–Capturing Algorithm for Compressible Multimaterial Elastic–Plastic Flows with Mie–Gr ¨uneisen Equation of State","authors":"Xiang Li, D. Ma, Nan-Sheng Liu null, Pei Wang","doi":"10.4208/aamm.oa-2021-0019","DOIUrl":"https://doi.org/10.4208/aamm.oa-2021-0019","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70494205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4208/aamm.oa-2021-0209
Min Li, Yumei Huang null, Y. Wen
{"title":"A Total Variation Based Method for Multivariate Time Series Segmentation","authors":"Min Li, Yumei Huang null, Y. Wen","doi":"10.4208/aamm.oa-2021-0209","DOIUrl":"https://doi.org/10.4208/aamm.oa-2021-0209","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70494483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4208/aamm.oa-2021-0363
Tingxiu Wang, Jie Zhou null, G. Hu
{"title":"An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory","authors":"Tingxiu Wang, Jie Zhou null, G. Hu","doi":"10.4208/aamm.oa-2021-0363","DOIUrl":"https://doi.org/10.4208/aamm.oa-2021-0363","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70494727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.4208/aamm.oa-2022-0188
Yuezheng Gong, Qi Hong, Chunwu Wang and Yushun Wang
{"title":"Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach","authors":"Yuezheng Gong, Qi Hong, Chunwu Wang and Yushun Wang","doi":"10.4208/aamm.oa-2022-0188","DOIUrl":"https://doi.org/10.4208/aamm.oa-2022-0188","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47692913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.4208/aamm.oa-2022-0247
Shitang Zhang
{"title":"Influence of the Radial Inertia Effect on the Propagation Law of Stress Waves in Thin-Walled Tubes","authors":"Shitang Zhang","doi":"10.4208/aamm.oa-2022-0247","DOIUrl":"https://doi.org/10.4208/aamm.oa-2022-0247","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42540878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.4208/aamm.oa-2022-0282
X. Liu, Zhiye Zhao, Nansheng Liu and Xiyun Lu
{"title":"Numerical Simulations of the Richtmyer–Meshkov Instability of Solid-Vacuum Interface","authors":"X. Liu, Zhiye Zhao, Nansheng Liu and Xiyun Lu","doi":"10.4208/aamm.oa-2022-0282","DOIUrl":"https://doi.org/10.4208/aamm.oa-2022-0282","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47678248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.4208/aamm.OA-2022-0340
Shijie Qin, S. Liao
The background numerical noise $varepsilon_{0} $ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $varepsilon(t)$ grows exponentially, say, $varepsilon(t) = varepsilon_{0} exp(kappa,t)$, where $kappa>0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in double precision, since the background numerical noise $varepsilon_{0}$ might stop decreasing because of the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $varepsilon_{0}$ to any required tiny level. A lot of successful applications show the novelty and validity of the CNS. In this paper, we further propose some strategies to greatly increase the computational efficiency of the CNS algorithms for chaotic dynamical systems. It is highly suggested to keep a balance between truncation error and round-off error and besides to progressively enlarge the background numerical noise $varepsilon_{0}$, since the exponentially increasing numerical noise $varepsilon(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.
{"title":"A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos","authors":"Shijie Qin, S. Liao","doi":"10.4208/aamm.OA-2022-0340","DOIUrl":"https://doi.org/10.4208/aamm.OA-2022-0340","url":null,"abstract":"The background numerical noise $varepsilon_{0} $ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $varepsilon(t)$ grows exponentially, say, $varepsilon(t) = varepsilon_{0} exp(kappa,t)$, where $kappa>0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in double precision, since the background numerical noise $varepsilon_{0}$ might stop decreasing because of the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $varepsilon_{0}$ to any required tiny level. A lot of successful applications show the novelty and validity of the CNS. In this paper, we further propose some strategies to greatly increase the computational efficiency of the CNS algorithms for chaotic dynamical systems. It is highly suggested to keep a balance between truncation error and round-off error and besides to progressively enlarge the background numerical noise $varepsilon_{0}$, since the exponentially increasing numerical noise $varepsilon(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42621923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-01DOI: 10.4208/aamm.oa-2022-0047
Ruo Li and Fanyi Yang
{"title":"A Reconstructed Discontinuous Approximation to Monge-Ampère Equation in Least Square Formulation","authors":"Ruo Li and Fanyi Yang","doi":"10.4208/aamm.oa-2022-0047","DOIUrl":"https://doi.org/10.4208/aamm.oa-2022-0047","url":null,"abstract":"","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45444585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-03DOI: 10.4208/aamm.OA-2022-0260
S. Liao
The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincar'{e}. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous ``small denominator problems'' never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely ``the method of directly defining inverse mapping'' (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous ``small denominator problems'' are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by ``small denominators''. The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called ``small denominators''.
{"title":"Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method","authors":"S. Liao","doi":"10.4208/aamm.OA-2022-0260","DOIUrl":"https://doi.org/10.4208/aamm.OA-2022-0260","url":null,"abstract":"The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincar'{e}. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous ``small denominator problems'' never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely ``the method of directly defining inverse mapping'' (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous ``small denominator problems'' are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by ``small denominators''. The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called ``small denominators''.","PeriodicalId":54384,"journal":{"name":"Advances in Applied Mathematics and Mechanics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46469524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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