Pseudospectral Methods in Applied Mathematics

（）CCV中的参考文献; [ ]研究参考文献。 该建议强调使用基于非经典正交网格[1]的伪谱方法，用于解决统计力学（4，9 -11），量子化学（1，7）和物理学（6），数学生物学和图像重建中的重要问题。伪谱方法是指偏微分方程和积分方程的数值解。(IE)在一个网格上[1]。普遍的选择是均匀傅立叶网格或非均匀切比雪夫求积。 傅立叶网格的缺点是吉布斯现象，这是傅立叶级数在跳跃不连续处的非谱收敛。 吉布斯现象污染了所有形式的断层扫描图像。一个正在进行的主要目标是开发一种识别振荡发生的“边缘”的方法，并采用以前的吉布斯分辨率技术[23，24]来解析图像。 同样，在均匀网格上插值数据会导致间隔两端的大振荡;龙格现象（RP）。RP可以解决与非均匀非经典求积谱精度。 数学生物学的主题包括用Fishers方程模拟的群体中基因的进展和用Fitzhugh-Nagumo方程研究的心脏组织中的螺旋波。本文用福克-普朗克方程（FPE）（4）和玻尔兹曼方程（BE）（1，11）的解研究了非平衡系统的平衡态逼近。 BE还用于模拟地球和火星大气层的高海拔地区，并与增强型极地外流探测器（ePOP）和MAVEN提供的卫星数据进行比较。断层扫描和磁共振图像的分辨率被认为是基于以前的工作的分辨率的吉布斯现象，分段光滑函数的傅立叶级数的非谱收敛。这些不同现象的建模基础是多维偏微分（PDE）和积分（IE）方程。费舍尔反应扩散方程[27]模拟了基因的进展，具有难以解决的局部行波和解。用二维Fitzhugh-Nagumo方程和不同的离子膜模型研究了被怀疑引起心律失常的心脏组织中的螺旋波[28]。对于等离子体和球状星团，BE近似与FPE与特定的扩散系数产生一个稳定的Kappa分布广泛用于空间科学。需要泊松求解器[15]来定义等离子体FPE中的漂移和扩散系数。 在流体动力学、等离子体物理和宇宙学等领域中，需要有足够的泊松方程求解器。需要有效和精确的数值方法来求解薛定谔方程（SE）（6）以描述许多化学和物理现象。

( ) 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.