The covariant derivative is a generalization of differentiating vectors. The Euclidean derivative is a special case of the covariant derivative in Euclidean space. The covariant derivative gathers broad attention, particularly when computing vector derivatives on curved surfaces and volumes in various applications. Covariant derivatives have been computed using the metric tensor from the analytically known curved axes. However, deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional (2D) surface. Consequently, computing the covariant derivative has been difficult or even impossible. A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface. A set of orthonormal vectors, known as moving frames, expand vectors to compute accurately covariant derivatives on 2D curved surfaces. The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols. The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames. Consequently, the proposed scheme can be extended to the general 2D curved surface. As an application, the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.
This paper proposes a new version of the high-resolution entropy-consistent (EC-Limited) flux for hyperbolic conservation laws based on a new minmod-type slope limiter. Firstly, we identify the numerical entropy production, a third-order differential term deduced from the previous work of Ismail and Roe [
The minimax path location problem is to find a path P in a graph G such that the maximum distance $d_G(v,P)$ from every vertex $v\in V(G)$ to the path P is minimized. It is a well-known NP-hard problem in network optimization. This paper studies the fixed-parameter solvability, that is, for a given graph G and an integer k, to decide whether there exists a path P in G such that $\mathop{\max}\limits_{v\in V(G)}d_G(v,P)\leqslant k$. If the answer is affirmative, then graph G is called k-path-eccentric. We show that this decision problem is NP-complete even for $k=1$. On the other hand, we characterize the family of 1-path-eccentric graphs, including the traceable, interval, split, permutation graphs and others. Furthermore, some polynomially solvable special graphs are discussed.
The present article mainly focuses on the fractional derivatives with an exponential kernel (“exponential fractional derivatives” for brevity). First, several extended integral transforms of the exponential fractional derivatives are proposed, including the Fourier transform and the Laplace transform. Then, the L2 discretisation for the exponential Caputo derivative with $\alpha \in (1,2)$ is established. The estimation of the truncation error and the properties of the coefficients are discussed. In addition, a numerical example is given to verify the correctness of the derived L2 discrete formula.
In this paper, finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed. The temporal derivative is in the Caputo-Hadamard sense for both cases. The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian. The schemes are proved to be unconditionally stable and convergent. The numerical results are in line with the theoretical analysis.
The family of Falk-Neilan $P_k$ finite elements, combined with the Argyris $P_{k+1}$ finite elements, solves the Reissner-Mindlin plate equation quasi-optimally and locking-free, on triangular meshes. The method is truly conforming or consistent in the sense that no projection/reduction is introduced. Theoretical proof and numerical confirmation are presented.