Britton Chang

A deterministic photon free method to solve radiation transfer equations

A new method to solve radiation transfer equations is presented. In the absence of scattering, material motion, and heat conduction, the photon variables can be eliminated from the fully implicit, multi-group, discrete-ordinate, finite difference (finite element) equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature. The solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived. The reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission. We solve the reduced set of equations by the Newton-GMRES method in which the Jacobian update is preconditioned by a linearization of this nonlinear heat equation. The performances of this new method and of the semi-implicit linear method, which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration, are compared on two test problems. The test results indicate that the new method can take larger time steps, requires less memory, is more accurate, and is competitive in speed with the semi-implicit linear method.

Spherical harmonic solutions of the Boltzmann transport equation via discrete ordinates

Abstract Discrete ordinates algorithms for deterministically simulating neutron transport via the Boltzmann transport equation are discussed. One difficulty these methods often have is the problem of ray effects. These undesirable numerical phenomena typically arise in problems with small scattering ratios and localized sources. A competing method that does not exhibit ray effects is the spherical harmonic, or P N , method in which particle fluxes are smoothly represented in a spherical harmonic basis. Although P N methods cannot have ray effects, the resulting system of equations in difficult to solve due to a denser coupling of terms than in discrete ordinates methods. A procedure, called harmonic projection , is presented that allows a P N solution to be obtained by solving only discrete ordinates equations. Thus, the quality of a P N solution can be obtained while exploiting the efficiency (and better parallelizability) of discrete ordinates methods. Numerical results on several benchmark problems are presented that demonstrate the effectiveness of the new algorithm in eliminating ray effects.

The incorporation of the semi-implicit linear equations into Newton’s method to solve radiation transfer equations

Abstract For large time steps, the nonlinear equations of radiation transfer may not be solved adequately by the semi-implicit linear approximation to yield physical solutions. This deficiency is rectified in three steps: the equations of the semi-implicit linear method are modified, the modified equations are incorporated into Newton’s method to solve nonlinear equations, and the transfer equations are solved by the resulting method. The new method also uses the Photon Free Method to search for the solution in a lower dimensional space than the space of the underlying transfer equations. Two algorithms are developed from the new method; they solve the modified semi-implicit linear equations by different approaches. The first is a physics approach; it solves the linear equations approximately by the Grey Transport Approximation. The second is a mathematical approach; it solves the linear equations exactly by the Sherman–Morrison–Woodbury formula of linear algebra. However, both algorithms yield the solution to the nonlinear system derived by the Photon Free Method. Therefore, their solutions are equal to within the specified tolerance of the nonlinear solver. Moreover, both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative. The time step which both methods can take is much larger than the time step in which time discretization error is discernible. We shall relate the mathematical approach to the Photon Free Method. Numerical results for three test problems are presented.

The conjugate gradient method solves the neutron transport equation h‐optimally

We report a serendipitous finding that the conjugate gradient method is able to obtain the solution to the linear system derived by the discontinuous Galerkin method for the discrete-ordinate approximation of the mono-energetic time-independent neutron transport equation to a specified accuracy in a number of iterations which is bounded independently of the order of the system. A theoretical explanation for this phenomenon is given. The results of an experiment to test the theory are presented. Copyright

Global error bounds for the Petrov–Galerkin discretization of the neutron transport equation

In this paper, we prove that the piecewise bilinear Petrov-Galerkin discretization for the mono-directional neutron transport equation described in (J. Comput. Phys. 1986; 64:96–111) is convergent and second-order accurate, provided that the true solution to the problem has continuous partial derivatives of all orders up through three. We do this by giving a bound on the 2-norm of the inverse of the system matrix that is independent of the mesh size. This shows that the global error is of the same order as the local truncation error. Copyright

Observation of nonsequential ionization of helium and its impact on intensity monitoring

We have measured the ion yields for helium and neon ionized by 120 femtosecond, 614 nanometer laser pulses with intensities up to 1016 watts per square centimeter. We have found that the He II and Ne II data exhibit features incompatible with standard nonresonant sequential ionization. These features reduce the usefulness of optical field ionization for monitoring laser intensity. For the experiment, we expect dynamic resonances to have little effect on the ionization, and we attribute the features to nonsequential ionization based on the simultaneous saturation of the features and the singly ionized charge states.

Basic features of coherent radiation generated by relativistic charge bunches

Abstract Radiation generated by relativistic charges can be analyzed and described in exquisite detail. One reason that such detailed analysis is possible is because the phases of radiated photons often are determined completely by the initial conditions of the relativistic charges and the radiating system. The phase relationships between the initial charges and the radiated photons represent coherence in the emitted radiation. A previous paper decribed how this coherence could affect the spatial and spectral distributions of radiation generated by a single charge in a periodic radiator. The present paper discusses a complementary issue; namely, how the temporal shape of a relativistic charge bunch can emphasize specific features of the radiation generated at a single interaction site.