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首页> 外文期刊>Applied Mathematical Modelling >Unconditionally stable high accuracy compact difference schemes for multi-space dimensional vibration problems with simply supported boundary conditions
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Unconditionally stable high accuracy compact difference schemes for multi-space dimensional vibration problems with simply supported boundary conditions

机译：具有简单支持边界条件的多维振动问题的无条件稳定高精度紧致差分格式

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Highlights•We have derived 2-level implicit methods for 4th order vibration problems.•The methods are compact and unconditionally stable.•We do not require to discretize the boundary conditions.•We require only a tri-diagonal solver to compute the proposed numerical methods.•We have solved 09 practical examples arising from physics.AbstractThe Euler–Bernoulli beam equation is a fourth order parabolic partial differential equation governing the transverse vibrations of a long and slender beam and is thus of interest in various engineering applications. In this study, we propose new two-level implicit difference formulas for the solution of vibration problem in one, two and three space dimensions subject to appropriate initial and boundary conditions. The proposed methods are fourth order accurate in space and second order accurate in time and are based upon a single compact stencil. The boundary conditions are incorporated in a natural way without any discretization or introduction of fictitious nodes. The derived methods are shown to be unconditionally stable for model linear problems. Some physical examples and their numerical results are given to illustrate the accuracy of the proposed methods. The test problems confirm that the computed solutions are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.
机译： 突出显示 • 我们已经导出了用于四阶振动问题的二级隐式方法。 • 方法紧凑且无条件稳定。 • 我们不需要离散化边界条件。 • 我们只需要一个三对角线求解器即可计算PR对数值方法。 • 我们已经解决了物理学中的09个实例。 摘要 Euler–Bernoulli梁方程是控制长而细长梁的横向振动的四阶抛物线偏微分方程，它是因此在各种工程应用中引起了人们的兴趣。在这项研究中，我们提出了新的两级隐式差分公式，用于在适当的初始和边界条件下解决一维，二维和三维空间中的振动问题。所提出的方法在空间上是四阶的，在时间上是二阶的，并且基于单个紧凑的模板。边界条件以自然方式合并，没有任何离散化或虚拟节点的引入。结果表明，对于模型线性问题，导出的方法是无条件稳定的。给出了一些物理算例及其数值结果，说明了所提方法的准确性。测试问题证实了计算出的解不仅与精确解非常吻合，而且还与早期研究中得出的解相称。

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