Numerical Methods For Transport Equations And Wave Propagations With Multiple Scales And Uncertainty

First,we develop a generalized polynomial chaos(gPC)based stochastic Galerkin(SG)for hyperbolic equations with random and singular coefficients.Due to the singular nature of the solution,the standard gPC-SG methods may suffer from a poor or even non convergence.Taking advantage of the fact that the discrete solution,by the central type finite difference or finite volume approximations in space and time for example,is smoother,we first discretize the equation by a smooth finite difference or finite volume scheme,and then use the gPC-SG approximation to the discrete system.The jump condition at the interface is treated using the immersed upwind methods introduced in[1,2].This yields a method that converges with the spectral accuracy for finite mesh size and time step.We use a linear hyperbolic equation with discontinuous and random coefficient,and the Liouville equation with discontinuous and random potential,to illustrate our idea,with both one and second order spatial discretizations.Spectral convergence is established for the first equation,and numerical examples for both equations show the desired accuracy of the method.Secondly,we study the stochastic Galerkin approximation for the linear transport equation with ran-dom inputs and diffusive scaling.We first establish uniform(in the Knudsen number)stability results in the random space for the transport equation with uncertain scattering coefficients,and then prove the uniform spectral convergence(and consequently the sharp stochastic Asymptotic-Preserving property)of the stochas-tic Galerkin method.A micro-macro decomposition based fully discrete scheme is adopted for the problem and proved to have a uniform stability.Numerical experiments are conducted to demonstrate the stability and asymptotic properties of the method.Thirdly,we propose a new time splitting Fourier spectral method for the semi-classical Schr?dinger equation with vector potentials.Compared with the results in[3],our method achieves spectral accuracy in space by interpolating the Fourier series via the NonUniform Fast Fourier Transform(NUFFT)algorithm in the convection step.The NUFFT algorithm helps maintain high spatial accuracy of Fourier method,and at the same time improve the efficiency from O(N~2)(of direct computation)to O(N log N)operations,where N is the total number of grid points.The kinetic step and potential step are solved by analytical solution with pseudo-spectral approximation,and,therefore,we obtain spectral accuracy in space for the whole method.We prove that the method is unconditionally stable,and we show improved error estimates for both the wave function and physical observables,which agree with the results in[4]for vanishing potential cases and are superior to those in[3].Extensive one and two dimensional numerical studies are presented to verify the properties of the proposed method,and simulations of 3D problems are demonstrated to show its potential for future practical applications.Finally,we investigate numerical approximations of the scalar conservation law with the Caputo deriva-tive,which introduces the memory effect.We construct the first order and the second order explicit upwind schemes for such equations,which are shown to be conditionally?~1contracting and TVD.However,the Ca-puto derivative leads to the modified CFL-type stability condition,(?t)~?=O(?x),where??(0,1]is the fractional exponent in the derivative.When?is small,such strong constraint makes the numerical im-plementation extremely impractical.We have then proposed the implicit upwind scheme to overcome this issue,which is proved to be unconditionally?~1contracting and TVD.Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.