Numerical cubature for fully symmetric regions and adaptive methods for PDEs

这项研究计划有两个主要目标。一个方向继续过去五年在单空间变量时变偏微分方程数值解方面的工作。微分方程出现在许多物理模型中，并且解通常表现为具有峰或急剧变化的窄层。在许多情况下，这些峰值或快速变化的区域随时间移动。对于一种能够捕捉这些特征的近似解方法，离散模型必须适应解。将开发软件来逼近具有可靠误差估计的解。我们将研究高阶插值在离线方法中的使用，以获得廉价和可靠的误差估计。使用等分布算法，我们将确保误差全局满足用户定义的容错。第二个重点，也是一个长期研究计划的延续，是建立具有特定对称性的多维区域积分数值计算方法。这些区域的例子有n维立方体、球面、单纯形、八面体、球面，以及具有对称权函数的整个n空间。如前面的一篇论文所示，球面和单纯形的公式之间有很强的联系。目的是研究八面体和单纯形的公式之间的联系。一个长期的计划是建立一个软件来接受该区域的对称性和所要求的多项式度作为输入，然后为公式计算一致的结构，建立定义方程，并求解它们。

This research programme has two principal thrusts. One direction continues work carried out in thelast five years on the numerical solution of time dependent partial differential equations in onespace variable. Differential equations occur in many physical models, and often the solutionexhibits narrow layers with peaks or sharp variations. In many cases these peaks or areas of rapidchange move in time. For an approximate method of solution to be able to catch these features, thediscrete model must adapt to the solution. Software to approximate to the solutions with reliable errorestimates will be developed. We will investigate the use of high order interpolants in a method oflines context, in order to obtain inexpensive and reliable error estimates. Using anequi-distribution algorithm we will ensure that the error globally satisfies a user-defined errortolerance.The second thrust, which also is a continuation of a long standing research programme, isthe construction of methods for the numerical evaluation of integrals over multidimensionalregions having specified symmetries.  Examples of such regions are the n dimensional cube,sphere, simplex, octahedron, spherical surface, and the whole of n-space with a symmetricweight function. As shown in an earlier paper there is a strong connection between formulasfor the surface of the sphere and for the simplex. It is intended to investigate connectionsbetween formulas for the octahedron and the simplex. A long term plan is to set up softwareto accept as input the symmetries of the region, and a requested polynomial degree, and thento compute consistent structures for the formulas, set up the defining equations, and solvethem.