The Euler equation of compressible flows is solved by the finite volume method, where high order accuracy is achieved by the reconstruction of each component of upwind fluxes of a flux splitting using the biased averaging procedure. Compared to the solution reconstruction in Godunov-type methods, its implementation is simple and easy, and the computational complexity is relatively low. This approach is parameter-free and requires neither a Riemann solver nor field-by-field decomposition. The numerical results from both dynamic and steady state calculations demonstrate the accuracy and robustness of this approach. Some techniques for the acceleration of the convergence to the steady state are discussed, including multigrid and multistage Runge–Kutta time methods.

We consider finite difference schemes based on the impulse density variable. We show that the original velocity—impulse density formulation of Oseledets is marginally ill-posed for the inviscid flow, and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity—impulse density formulation. We present numerical evidence of this instability. We then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation. Finally we give a simplified velocity—impulse density formulation which is free of these complications and yet retains the nice features of the original velocity—impulse density formulation with regard to the treatment of boundary. We present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non-staggered grids.

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively

Recent technical advances lead to the coupling of PET and MRI scanners, enabling one to acquire functional and anatomical data simultaneously. In this paper, we propose a tight frame based PET-MRI joint reconstruction model via the joint sparsity of tight frame coefficients. In addition, a nonconvex balanced approach is adopted to take the different regularities of PET and MRI images into account. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm is proposed, and the global convergence is present based on the Kurdyka--Łojasiewicz property. Finally, the numerical experiments show that our proposed models achieve better performance over the existing PET-MRI joint reconstruction models.

We develop a PetrovGalerkin stabilization method for multiscale convectiondiffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not be sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems and thus, develop a systematic approach for finding reduced-order approximations of the solution. We start from a PetrovGalerkin framework using optimal weighting functions. We introduce an auxiliary variable to a mixed formulation of the problem. The auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a dimensionally reduced space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full