Computational algorithms for non-linear discontinuous problems

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear partial differential equation with a discontinuous solution, Fourier spectral method produces poor pointwise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this part of the dissertation, we assess the accuracy of Fourier spectral method applied to nonlinear hyperbolic conservation laws through a careful numerical case study. We find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials.; When truncated Fourier series is used to approximate a periodic function, the available informations are contained in the first few Fourier coefficients. The rate of decay of the Fourier modes is dictated by smoothness of the underlying function. The smoother the function is, the faster the Fourier modes decay. If the function is discontinuous, its Fourier modes decay only linearly. Hence, the first few modes contain very limited information of the function. In this part, we address the question whether the jump location of a periodic function with a single discontinuity can be accurately found from a few Fourier modes. We found that indeed this information is present, and we propose an algorithm to extract it.; In the last part of this thesis, we tackle the problem of code optimization through implementation of a high-order essentially non-oscillatory (ENO) finite-difference scheme to solve the two-dimensional compressible Euler equations on an Intel Paragon parallel supercomputer. Practical issues such as vectorization and parallel efficiency are discussed. High performance are demonstrated by some numerical testing problems.