A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts (upwind gSBP) schemes in space and implicit-explicit Runge-Kutta (IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly, while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first- and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin (DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay (BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space, since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.
Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an dual complex matrix is diagonalizable if and only if it has exactly n eigenvalues with n appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.
In this paper, the efficient preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks. With the aid of the singular value decomposition technique, the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks. Moreover, the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case. On account of the favorable spectral properties of the PMHSS preconditioned matrix, a parameter free Chebyshev accelerated PMHSS (CAPMHSS) method is established to further improve its convergence rate. Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.
In this paper, a linearized energy-stable scalar auxiliary variable (SAV) Galerkin scheme is investigated for a two-dimensional nonlinear wave equation and the unconditional superconvergence error estimates are obtained without any certain time-step restrictions. The key to the analysis is to derive the boundedness of the numerical solution in the -norm, which is different from the temporal-spatial error splitting approach used in the previous literature. Meanwhile, numerical results are provided to confirm the theoretical findings.
This paper is concerned with the reflected stochastic Burgers equation driven both by the Brownian motion and by the Poisson random measure. The existence and uniqueness of solutions are established. The penalization method plays an important role.
In this paper, the modulus-based matrix splitting (MMS) iteration method is extended to solve the horizontal quasi-complementarity problem (HQCP), which is characterized by the presence of two system matrices and two nonlinear functions. Based on the specific matrix splitting of the system matrices, a series of MMS relaxation iteration methods are presented. Convergence analyses of the MMS iteration method are carefully studied when the system matrices are positive definite matrices and -matrices, respectively. Finally, two numerical examples are given to illustrate the efficiency of the proposed MMS iteration methods.
In this paper, the existence theorem for a quasi solution of an inverse fractional stochastic parabolic equation driven by multiplicative noise in the form is given. In this equation, the fractional derivative is considered in the Caputo sense. Also, the random function g is unknown and should be determined. To identify the unknown coefficient, the minimization and stochastic variational formulation methods in a fractional stochastic Sobolev space are used. Indeed, we obtain a stability estimation and then prove the continuity of the minimization functional using obtained stability estimation. These results show the existence of the quasi solution for the mentioned problem.
In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.
In this paper, an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient. Since the solution of such an equation in general has a typical weak singularity near the initial time , the time-fractional derivative with order in (0, 1) is discretized by L2-1σ formula on nonuniform meshes. For the spatial derivative, the local discontinuous Galerkin (LDG) method is employed. A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality. Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.