We present a finite-volume formulation for the lattice Boltzmann method (FVLBM) based on standard bilinear quadrilateral elements in two dimensions. The accuracy of this scheme is demonstrated by comparing the velocity field with the analytical solution of the Navier-Stokes equations for time dependent rotating Couette flow and Taylor vortex flow. To demonstrate the flexibility of the scheme, w...

In this talk we discuss the problem of constructing effective high order methods for the numerical solution of hyperbolic systems of balance laws, in presence of stiff source. Because of the stiffness, the use of implicit integrators is advisable, so that no restrictions on the time step due to small relaxation time will appear. Two different relaxation systems will be considered, namely hyperb...

We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocit...

The numerical solution of parametrized partial differential equations (P2DEs ) can be a very time-consuming task if many parameter constellations have to be simulated by high-resolution schemes. Such scenarios may occur in parameter studies, optimization, control, inverse problems or statistical analysis of a given P2DE . Reduced Basis (RB) methods allow to produce fast reduced models that are ...

Pressure driven microscale gas flow-separation has been investigated by solving the compressible Navier-Stokes (NS) system of equations. A two dimensional explicit finite volume (FV) compressible flow solver has been developed using modified advection upwind splitting methods (AUSM+) with no-slip/first order Maxwell’s velocity slip conditions to predict the flowseparation behavior in microdimen...

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with d...

In this paper, we consider an implicit finite difference method for solving fuzzy partial differential equations (FPDEs). We present stability of this method and solve the parabolic equation with this scheme.

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general H1 initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in H1 towards a dissipative weak solution of C...