Volume 10, Issue 1

A novel approach to a discrete Hessian recovery is suggested. Local convergence of the discrete Hessian towards the differential Hessian is established, for a class of adaptive meshes.

In this paper, we consider solutions of Fredholm integral equations of the second kind where the kernel functions are asymptotically smooth or products of such functions with highly oscillatory coefficient functions. We present a scheme based on polynomial interpolation to approximate matrices A from the discretization of these integral operators. Our approximation matrix B is obtained by partitioning the domain on which the kernel function is defined into subdomains of different sizes and approximating the kernel function at each subdomain by interpolation polynomial at the Chebyshev points. Although B is dense, it can still be constructed in O ( nk ) operations, requires O ( nk ) storage and the product B y can be obtained in O ( nk log n )operations, where n is the size of the matrix and k is the degree of the interpolation polynomial used. We prove that the Frobenius norm ‖ A – B ‖ F ⩽ ε if k is of O (log ε –1 ) for smooth kernels (including log| x – t |) and of O (loglog n + log ε –1 ) for weakly singular kernels such as | x – t | –1/2 . Comparison with the wavelet-like method by Alpert et al. [SIAM J. Sci. Comp 14: 159–184, 1993] shows that our method requires less memory and is more accurate.

In this paper we compare the accuracy and efficiency of a finite volume scheme solving the ideal MHD equations in one and two space dimensions for interface fluxes obtained by six different MHD Riemann solvers. We suggest a new solver of HLLEM-type (MHD–HLLEM) and show that it is the cheapest one in our comparison. Moreover, we study if the extra effort required for higher order schemes is justified by the gain in accuracy. The two-dimensional version of our code is based on unstructured triangular meshes.