The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by ...The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.展开更多
ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lew...ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.展开更多
Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the e...Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations vere also given. For the Cauchy problem on the hyper-plane {t = O}, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.展开更多
This paper continues discussing the problems of numerically solving the shallow water circulation on the basis of ref. 1, For the numerical method proposed in ref. 1, we applied a storage method with dense matrices, w...This paper continues discussing the problems of numerically solving the shallow water circulation on the basis of ref. 1, For the numerical method proposed in ref. 1, we applied a storage method with dense matrices, which abandoned usual bandwidth concept and attained the intention of saving interior storage, computing time and amount of preparing work before computing. The circulation considered the effect of small islands was successfully simulated by specially dealing with the bottom friction terms and the boundary conditions. In addition, we discussed the action of bottom friction on the dissipation of tidal energy and its effect on stability of period motion.展开更多
开发了一种基于B样条的高阶面元法用来求解浅水船舶兴波问题。船体表面和自由面上分别布置Rankine源,同时利用镜像原理来计及水底的影响。物体几何用 B 样条曲面精确表示。在求得边界面上的源强密度分布后,物面上的速度势用B样条来表示...开发了一种基于B样条的高阶面元法用来求解浅水船舶兴波问题。船体表面和自由面上分别布置Rankine源,同时利用镜像原理来计及水底的影响。物体几何用 B 样条曲面精确表示。在求得边界面上的源强密度分布后,物面上的速度势用B样条来表示。数值计算中采用配置方法,并且用高斯—勒让德公式来计算方程中的积分。为了验证文中方法的有效性,用本方法计算了Wigley船在深水和浅水中的兴波水动力和波形,所得数值结果与试验结果和其它数值结果进行了比较,吻合程度令人满意,表明本方法被用来求解浅水船舶兴波问题是有效的。展开更多
基金supported by NSFC (10771074)NSFC-NSAF(10976026)+1 种基金Yang was partially supported by NSFC (10801055 10901057)
文摘The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.
文摘ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.
基金Project supported by the National Natural Science Foundation of China (Grant No: 90411006, 40175014) and the Key Foundation of Shanghai Municipal Commission of Science and Technology (Grant No: 02DJ14032)
文摘Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations vere also given. For the Cauchy problem on the hyper-plane {t = O}, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.
文摘This paper continues discussing the problems of numerically solving the shallow water circulation on the basis of ref. 1, For the numerical method proposed in ref. 1, we applied a storage method with dense matrices, which abandoned usual bandwidth concept and attained the intention of saving interior storage, computing time and amount of preparing work before computing. The circulation considered the effect of small islands was successfully simulated by specially dealing with the bottom friction terms and the boundary conditions. In addition, we discussed the action of bottom friction on the dissipation of tidal energy and its effect on stability of period motion.
基金Supported by the National Natural Science Foundation of China (Grant No: 10572094)the Natural Science Foundation of Shanghai (Grant No: 06ZR14050)
文摘开发了一种基于B样条的高阶面元法用来求解浅水船舶兴波问题。船体表面和自由面上分别布置Rankine源,同时利用镜像原理来计及水底的影响。物体几何用 B 样条曲面精确表示。在求得边界面上的源强密度分布后,物面上的速度势用B样条来表示。数值计算中采用配置方法,并且用高斯—勒让德公式来计算方程中的积分。为了验证文中方法的有效性,用本方法计算了Wigley船在深水和浅水中的兴波水动力和波形,所得数值结果与试验结果和其它数值结果进行了比较,吻合程度令人满意,表明本方法被用来求解浅水船舶兴波问题是有效的。
