Pseudospectral Methods in Applied Mathematics

()参考CCV；[]研究参考。 这一建议强调使用基于非经典正交网格的伪谱方法[1]来解决统计力学(4，9-11)、量子化学(1，7)和物理(6)、数学生物学和图像重建中的重要问题。拟谱方法指的是偏微分方程组和积分方程组的数值解。(即)在网格[1]上。常见的选择是均匀傅里叶网格或非均匀切比雪夫求积。傅里叶网格的缺点是Gibbs现象，这是函数在跳跃间断处的傅里叶级数的非谱收敛。吉布斯现象污染了所有形式的断层成像中的图像。一个正在进行的主要目标是开发一种方法来识别发生振荡的“边缘”，并使用以前的Gibbs分辨率技术[23，24]来解析图像。同样，数据在均匀网格上的内插会导致区间末端的大振荡，即Runge现象(Runge现象)。用非均匀非经典求积法可求解谱精度较高的反问题。 数学生物学的主题包括用Fishers方程模拟的种群中基因的进展，以及用Fitzhugh-Nagumo方程研究的心脏组织中的螺旋波。利用Fokker-Planck方程(FPE)(4)和Boltzmann方程(BE)(1，11)的解，研究了非平衡系统的平衡方法。BE还被用来模拟地球和火星大气的高海拔地区，并与来自增强型极地流出探测器(EPOP)和MAVEN的现有卫星数据进行比较。层析成像和磁共振成像图像的分辨率是在以前解决吉布斯现象的基础上考虑的，吉布斯现象是分段光滑函数的傅里叶级数的非谱收敛。模拟这些不同现象的基础是多维偏微分方程组和积分方程组。费舍尔反应扩散方程[27]模拟了一个基因的进展，它有局部化的行波和难以解析的解。用二维Fitzhugh-Nagumo方程和不同的离子膜模型研究了心脏组织中可能引起心律失常的螺旋波[28]。对于等离子体和球状星团，BE近似为FPE，它具有特定的扩散系数，产生在空间科学中广泛使用的稳定的Kappa分布。需要泊松解算器[15]来定义等离子体FPE中的漂移和扩散系数。泊松方程的高效泊松解在流体力学、等离子体物理和宇宙学中都是需要的。要求解薛定谔方程(SE)(6)来描述许多化学和物理现象，需要高效而准确的数值方法。

( ) Refs in CCV; [ ] Research Refs. This proposal emphasizes the use of pseudospectral methods based on nonclassical quadrature grids [1] applied to the solution of important problems in statistical mechanics (4,9-11), quantum chemistry (1,7) and physics (6), mathematical biology and image reconstruction. A pseudospectral method refers to a numerical solution of PDEs and Integral eqs. (IE) on a grid [1]. The popular choices are the uniform Fourier grid or the nonuniform Chebyshev quadrature. The disadvantage of a Fourier grid is the Gibbs phenomena which is the non-spectral convergence of the Fourier series for functions at jump discontinuties. The Gibbs phenomenon contaminates images in all forms of tomography. A major ongoing objective is to develop a method of identifying the "edges" where the oscillations occur and to employ the previous Gibbs resolution techniques [23,24] to resolve the images. Likewise, the interpolation of data on a uniform grid leads to large oscillations at the ends of the interval; the Runge phenomenon (RP). The RP can be resolved with nonuniform nonclassical quadratures with spectral accuracy. The topics in mathematical biology include the advance of a gene in a population modeled with Fishers equation and spiral waves in cardiac tissue studied with the Fitzhugh-Nagumo equation. The approach to equilibrium for nonequilibrium systems are studied with solutions of the Fokker Planck equation (FPE) (4) and the Boltzmann equation (BE) (1,11). The BE is also used to model the the high altitude regions of the atmospheres of Earth and Mars in comparison with available satellite data from enhanced Polar Outflow Probe (ePOP) and MAVEN. The resolution of images for tomography and magnetic resonance images is considered based on previous work for the resolution of the Gibbs phenomenon, the nonspectral convergence of a Fourier series of piecewise smooth functions. The basis for the modeling these different phenomena are multidimensional partial differential (PDE) and integral (IE) equations. Fishers reaction-diffusion equation [27], which models the advance of a gene, has localized traveling waves and solutions that are difficult to resolve. Spiral waves in cardiac tissue that are suspected to give rise to cardiac arrhythmias [28] are studied with the two-dimensional Fitzhugh-Nagumo equation with different ionic membrane models. For plasmas and globular clusters, the BE is approximated with a FPE which with particular diffusion coefficients yields a steady Kappa distribution used extensively in space science. A Poisson solver [15] is required to define the drift and diffusion coefficients in the FPE for plasmas. Ffficient Poisson solvers for Poisson's equation are required in fluid dynamics, plasma physics and cosmology. Efficient and accurate numerical methods are required to solve the Schroedinger equation (SE) (6) to describe many chemical and physical phenomenon.