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首页> 外文期刊>International Journal for Numerical Methods in Fluids >A PIECEWISE-LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO-POINT BOUNDARY-VALUE PROBLEMS
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A PIECEWISE-LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO-POINT BOUNDARY-VALUE PROBLEMS

机译：差分方程的分段线性化方法：两点边值问题

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摘要

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton-Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.
机译：提出了求解常微分方程两点边值问题的分段线性化方法。这些问题可以用具有解析解的分段线性问题来近似，并简化为在左边界找到解的斜率，从而满足区间右端的边界条件。这导致相当复杂的非线性代数方程组，该系统可以简化为单个非线性方程组，其未知数是区间左边界处的解的斜率，并且可以通过牛顿来获得其解。 -拉夫森法。这等效于使用分段线性化技术和射击方法将边界值问题解决为初始值。结果表明，对于以线性算子为特征的问题，可以采用基于叠加原理和分段线性化方法的技术。对于这些问题，分段线性化方法的准确性是二阶的。还表明，对于线性问题，分段线性化方法的精度优于四阶精确技术。对于本文所考虑的线性奇异摄动问题，整体分段线性化的精度高于有限差分法和有限元方法。对于非线性问题，分段线性化方法的精度在大多数情况下低于四阶方法，但由于非线性项的线性化，可以与二阶技术相比。

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