This paper develops the high-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

In this paper, we analyse the stochastic collocation method for a linear Schrödinger equation with random inputs, where the randomness appears in the potential and initial data and is assumed to be dependent on a random variable. We focus on the convergence rate with respect to the number of collocation points. Based on the interpolation theories, the convergence rate depends on the regularity of the solution with respect to the random variable. Hence, we investigate the dependence of the stochastic regularity of the solution on that of the random potential and initial data. We provide sufficient conditions on the random potential and initial data to ensure the smoothness of the solution and the spectral convergence. Finally, numerical results are presented to support our analysis.

In this paper, we design a collocation method to solve the fractional Ginzburg-Landau equation. A Jacobi collocation method is developed and implemented in two steps. First, we space-discretize the equation by the Jacobi-Gauss-Lobatto collocation (JGLC) method in one- and two-dimensional space. The equation is then converted to a system of ordinary differential equations (ODEs) with the time variable based on JGLC. The second step applies the Jacobi-Gauss-Radau collocation (JGRC) method for the time discretization. Finally, we give a theoretical proof of convergence of this Jacobi collocation method and some numerical results showing the proposed scheme is an effective and high-precision algorithm.

In this paper, an initial boundary value problem of the space-time fractional diffusion equation is studied. Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods. And then based on the discretization scheme, reliable a posteriori error estimates for the spectral approximation are derived. Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.

In this paper, we study the two-dimensional Helmholtz scattering problem by a locally perturbed line with impedance boundary condition. Different from the problem with Dirichlet boundary condition, the Green function of the Helmholtz equation with impedance boundary condition becomes very complicated and comprises surface waves along the locally perturbed line. A uniaxial perfectly matched layer (UPML) method is proposed to truncate the half plane into a bounded computational domain.  The main contribution of this paper is to prove the well-posedness of the PML problem and the exponential convergence of the approximate solution to the exact solution as either the thickness or the medium parameter of PML increases.

In this paper, a mixed finite element method is investigated for the Maxwell's equations in Debye medium with a thermal effect. In particular, in two dimensional case, the zero order Nédélec element $(Q_{01}\times Q_{10})$, the piecewise constant space $Q_0$ element, and the bilinear element $Q_{11}$ are used to approximate the electric field E and the polarization electric field P, the magnetic field H, and the temperature field $u$, respectively. With the help of the high accuracy results, mean-value technique and interpolation postprocessing approach, the convergent rate $\mathcal{O}(\tau+h^2)$ for global superconvergence results are obtained under the time step constraint $\tau=\mathcal{O}(h^{1+\gamma}),$ $ \gamma>0$ by using the linearized backward $Euler$ finite element discrete scheme. At last, a numerical experiment is given to verify the theoretical analysis and the validity of our method.

In this paper, we solve linear parabolic integral differential equations using the weak Galerkin finite element method (WG) by adding a stabilizer. The semi-discrete and fully-discrete weak Galerkin finite element schemes are constructed. Optimal convergent orders of the solution of the WG in $L^2$ and $H^1$ norm are derived. Several computational results confirm the correctness and efficiency of the method.

In this paper, the discontinuous Galerkin method is applied to solve the multi-pantograph delay differential equations. We analyze the optimal global convergence and local superconvergence for smooth solutions under uniform meshes. Due to the initial singularity of the forcing term $f$, solutions of multi-pantograph delay differential equations are singular. We obtain the relevant global convergence and local superconvergence for weakly singular solutions under graded meshes. The numerical examples are provided to illustrate our theoretical results.

We propose a robust approximate solver for the hydro-elastoplastic solid material, a general constitutive law extensively applied in explosion and high speed impact dynamics, and provide a natural transformation between the fluid and solid in the case of phase transitions. The hydrostatic components of the solid is described by a family of general Mie-Grüneisen equation of state (EOS), while the deviatoric component includes the elastic phase, linearly hardened plastic phase and fluid phase. The approximate solver provides the interface stress and normal velocity by an iterative method. The well-posedness and convergence of our solver are proved with mild assumptions on the equations of state. The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds. Several numerical examples, including Riemann problems, shock-bubble interactions, implosions and high speed impact applications, are presented to validate the approximate solver.

In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term $\theta\Delta  \mathbf{u}$ in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter $\nu$, by choosing suitable parameter $\theta$. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition $\Delta\leq k_0$, $k_0>0$, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement. Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement. The mixed method is locking-free, i.e., the convergence does not deteriorate in the nearly incompressible or incompressible case. The optimal convergence order is shown in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement, respectively. Finally, some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.