Discrete numerical approximations with additional conserved quantities are developed here both for barotropic geophysical flows generalizing the 2D incompressible fluid equations and suitable discretizations of the Burgers-Hopf equation. Mathematical, numerical, and statistical properties of these approximations are studied below in various different contexts through the symbiotic interaction of mathematical theory and scientific computing. The new contributions include an explicit concrete discussion of the sine-bracket spectral truncation with many conserved quantities for 2D incompressible flow, a theoretical and numerical comparison with the standard spectral truncation, and a rigorous proof of convergence to suitable weak solutions in the limit as the number of Fourier modes increases. Systematic discretizations of the Burgers-Hopf equation are developed, which conserve linear momentum and a non-linear energy; careful numerical experiments regarding the statistical behavior of these models indicate that they are ergodic and strongly mixing but do not have equipartition of energy in Fourier space. Furthermore, the probability distribution of the values at a single grid point can be highly non-Gaussian with two strong isolated peaks in this distribution. This contrasts with earlier results for statistical behavior of difference schemes which conserve a quadratic energy. The issues of statistically relevant conserved quantities are introduced through a new case study for the Galerkin-truncated Burgers-Hopf model.

We study the asymptotics of front speeds of the reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity and zero mean stationary ergodic Gaussian shear advection on the entire plane. By exploiting connections of KPP front speeds with the almost sure Lyapunov exponents of the parabolic Anderson problem, and with the homogenized Hamiltonians of Hamilton-Jacobi equations, we show that front speeds enhancement is quadratic in the small root mean square (rms) amplitudes of white in time zero mean Gaussian shears, and it grows at the order of the large rms amplitudes. However, front speeds diverge logarithmically if the shears are time independent zero mean stationary ergodic Gaussian fields.

The weighted minimal surface problem in heterogeneous media is studied in this paper. The solution to the weighted minimal surface problem is continuous but the derivatives have a jump across the interface where the medium property is discontinuous. The jump condition of the derivatives derived in this paper generalized the Snell's law in geometric optics to weighted minimal surfaces of co-dimension one in any dimensional space. A numerical method based on the gradient flow and the maximum principal preserving immersed interface method is developed to solve this nonlinear elliptic interface problem with jump conditions. Numerical computations are presented to verify both the analysis and the numerical algorithm.

We prove the existence of global entropy solutions in L^infinity to the multidimensional Euler equations and Euler-Poisson equations for compressible isothermal fluids with spherically symmetric initial data that allows vacuum and unbounded velocity outside a solid ball. The multidimensional existence problem can be reduced to the existence problem for the one-dimensional Euler equations and Euler-Poisson equations with geometrical source terms. Due to the presence of the geometrical source terms, new variables- weighted density and momentum-are first introduced to transform the nonlinear system into a new nonlinear hyperbolic system to reduce the geometric source effect. We then develop a shock capturing scheme of Lax-Friedrichs type to construct approximate solutions for the weighted density and momentum. Since the velocity may be unbounded, the Courant-Friedrichs-Lewy stability condition may fail for the standard fractional-step Lax-Friedrichs scheme; hence we introduce a cut-off technique to modify the approximate density functions and adjust the ratio of the space and time mesh sizes to construct our approximate solutions. Finally we establish the convergence and consistency of the approximate solutions using the method of compensated compactness and obtain global entropy solutions in L^infinity. The solutions we obtain allow unbounded velocity near vacuum, one of the essential difficulties here, which is different from the isentropic case.

After spending a quarter at the UCLA in the early 1998, I got to know Professor Stanley Osher better. He always seems to have insights and nice ideas to do computations for complicated problems. Thus at those times when I encounter problems for which analysis seem either practically impossible or extremely difficult and, for which some reliable computations may give either a reasonable solution or some hints, I often turn to experts like Stanley to see if they can do anything about them. The present article is of such nature, and I would like to dedicate it to Stanley on the occasion of his 60th birthday. I had some ideas of handling the backward parabolic problem about one year ago, then I learned from P. Lax some earlier works (nearly 50 years ago!) of F. John, see [1]. Though the approach I had is apparently rather different from that of F. John, they seem to have some deep connections. I shall explain some notions (which I found rather amusing) introduced in John's work in the next section. A solution to the problem (*) (or a somewhat more general problem) will be explained in section 3. In the final section, I shall describe a few issues which may be of interest from both theory and computations. I have no intention here to make various statements or estimates more refined. The goal here is to present the problems and certain point views on such problems. I wish to thank P. Lax for bringing John's work to my attention and for several interesting discussions.

This is a continuation of our work [KG], [GGK], where we studied a gradient (flow) system of an energy whose energy density is not C1 so that the diffusivity in the equation is very strong and its effect is even nonlocal. In this paper we consider the case when the values of unknowns are constrained. To be specific we consider a gradient (flow) system of the total variations of mappings with constraint of their values. We also study the behavior of solution when M is the unit circle S1. The equation of the motion of the plateau is presented, which is written in the form of reducing ODE.We identify the form of stationary solutions and prove that the solution becomes a stationary solution in finite time. In the last part of this paper we demonstrate numerical simulations of the S1-valued problem. Although our theoretical approach is restricted to the piecewise constant solutions at this point, our method also applies to calculating the evolution of the solutions in more general class. Our method to solve S1-valued problem employs an angle variable, thus it is different from the method in [MSO] whereM is represented as a zero level set of some functions.

We describe the generic solution of the Riemann problem near a point of resonance in a general 2x2 system of balance laws coupled to a stationary source. The source is treated as a conserved quantity in an augmented 3x3 system, and Resonance is between a nonlinear wave family and the stationary source. Transonic compressible Euler flow in a variable area duct, as well as spherically symmetric flow, are shown to be special cases of the general class of equations studied here.

In this paper we study a class of one-sided post-processing techniques to enhance the accuracy of the discontinuous Galerkin methods. The applications considered in this paper are linear hyperbolic equations, however the technique can be used for the solution to a discontinuous Galerkin method solving other types of partial differential equations, or more general approximations, as long as there is a higher order negative norm error estimate for the numerical solution. The advantage of the one-sided post-processing is that it uses information only from one side, hence it can be applied up to domain boundaries, a discontinuity in the solution, or an interface of different mesh sizes. This technique allows us to obtain an improvement in the order of accuracy from k+1 of the discontinuous Galerkin method to 2k+1 of the post-processed solution, using piecewise polynomials of degree k, throughout the entire domain and not just away from the boundaries, discontinuities, or interfaces of different mesh sizes.

Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and force by solving the governing Poisson-Boltzmann equation. This paper focuses on the following issues: (i) the point charge singularity problem, (ii) the dielectric discontinuity problem across a molecular surface, and (iii) the infinite domain problem. Green's function associated with the point charges plus a harmonic function is introduced as the zeroth order approximation to the solution to solve the point charge singularity problem. A jump condition capturing finite difference scheme is adopted to solve the discontinuity problem across molecule surfaces, where a body-fitting grid is used. The infinite domain problem is solved by mapping the outer infinite domain into a finite domain. The corresponding stiffness matrix is symmetric and positive definite, therefore, fast algorithm such as preconditioned conjugate gradient method can be applied for inner iteration. Finally, the resulting scheme is second order accurate for both the potential and its gradient.

We consider a vibrating system with piezoelectric patches whose minimum vibration frequency is to be minimized subject to some constraint on the patch geometry. A numerical scheme is constructed using the level set method devised in [12] and the projected gradient method devised in [14] to optimize the patch geometry. An integral equation approach introduced in [4, 17, 20, 21] is also presented for the computation of the corresponding eigenvalue problem.