Martin Frank

Kerstin Küpper, Martin Frank, Shi Jin

We propose a two-dimensional asymptotic preserving scheme for linear transport equations with diffusive scalings. It is an extension of the time splitting developed by Jin, Pareschi and Toscani [SINUM,2000], but uses spatial discretizations on staggered grids, which preserves the discrete diffusion limit with a more compact stencil. The first novelty of this paper is that we propose a staggering in two dimensions that requires fewer unknowns than one could have naively expected. The second contribution of this paper is that we rigorously analyze the scheme of Jin, Pareschi, and Toscani [SINUM,2000] We show that the scheme is AP and obtain an explicit CFL condition, which couples a hyperbolic and a parabolic condition. This type of condition is common for asymptotic preserving schemes and guarantees uniform stability with respect to the mean free path. In addition, we obtain an upper bound on the relaxation parameter, which is the crucial parameter of the used time discretization. Several numerical examples are provided to verify the accuracy and asymptotic property of the scheme.

Benjamin Seibold, Martin Frank

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.

Richard Barnard, Martin Frank, Michael Herty

We study the problem of finding an optimal radiotherapy treatment plan. A time-dependent Boltzmann particle transport model is used to model the interaction between radiative particles with tissue. This model allows for the modeling of inhomogeneities in the body and allows for anisotropic sources modeling distributed radiation---as in brachytherapy---and external beam sources---as in teletherapy. We study two optimization problems: minimizing the deviation from a spatially-dependent prescribed dose through a quadratic tracking functional; and minimizing the survival of tumor cells through the use of the linear-quadratic model of radiobiological cell response. For each problem, we derive the optimality systems. In order to solve the state and adjoint equations, we use the minimum entropy approximation; the advantages of this method are discussed. Numerical results are then presented.

Graham W. Alldredge, Martin Frank, Cory D. Hauck

We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We show that this equation still retains many of the properties of the original equations, including hyperbolicity, an entropy-dissipation law, and rotational invariance. The cost of the regularization is mismatch between the moment vector of the solution and that of the ansatz returned by the regularized optimization problem. However, we show how to control this error using the parameter defining the regularization. This suggests that with proper choice of the regularization parameter, the new method can be used to generate accurate solutions of the original entropy-based moment method, and we confirm this with numerical simulations.

Thomas Camminady, Martin Frank, Kerstin Küpper et al.

Solving the radiation transport equation is a challenging task, due to the high dimensionality of the solution's phase space. The commonly used discrete ordinates (S$_N$) method suffers from ray effects which result from a break in rotational symmetry from the finite set of directions chosen by S$_N$. The spherical harmonics (P$_N$) equations, on the other hand, preserve rotational symmetry, but can produce negative particle densities. The discrete ordinates (S$_N$) method, in turn, by construction ensures non-negative particle densities. In this paper we present a modified version of the S$_N$ method, the rotated S$_N$ (rS$_N$) method. Compared to S$_N$, we add a rotation and interpolation step for the angular quadrature points and the respective function values after every time step. Thereby, the number of directions on which the solution evolves is effectively increased and ray effects are mitigated. Solution values on rotated ordinates are computed by an interpolation step. Implementation details are provided and in our experiments the rotation/interpolation step only adds 5% to 10% to the runtime of the S$_N$ method. We apply the rS$_N$ method to the line-source and a lattice test case, both being prone to ray-effects. Ray effects are reduced significantly, even for small numbers of quadrature points. The rS$_N$ method yields qualitatively similar solutions to the S$_N$ method with less than a third of the number of quadrature points, both for the line-source and the lattice problem. The code used to produce our results is freely available and can be downloaded.

Martin Frank, Cory D. Hauck, Edgar Olbrant

We derive a hierarchy of closures based on perturbations of well-known entropy-based closures; we therefore refer to them as perturbed entropy-based models. Our derivation reveals final equations containing an additional convective and diffusive term which are added to the flux term of the standard closure. We present numerical simulations for the simplest member of the hierarchy, the perturbed M1 or PM1 model, in one spatial dimension. Simulations are performed using a Runge-Kutta discontinuous Galerkin method with special limiters that guarantee the realizability of the moment variables and the positivity of the material temperature. Improvements to the standard M1 model are observed in cases where unphysical shocks develop in the M1 model.

Martin Frank, Benjamin Seibold

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.

M. Paul Laiu, Martin Frank, Cory D. Hauck

We present a positive and asymptotic preserving numerical scheme for solving linear kinetic, transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and realizability limiters in the angular discretization. Under mild assumptions on the initial condition and time step, the scheme becomes a consistent numerical discretization for the limiting diffusion equation when the scattering cross-section tends to infinity. The scheme also preserves positivity of the particle concentration on the space-time mesh and therefore fixes a common defect of spectral angular discretizations. The scheme is tested on the well-known line source benchmark problem with the usual uniform material medium as well as a medium composed from different materials that are arranged in a checkerboard pattern. We also report the observed order of space-time accuracy of the proposed scheme.

Philipp Otte, Martin Frank

We derive Lattice Boltzmann (LBM) schemes to solve the Linearized Euler Equations in 1D, 2D, and 3D with the future goal of coupling them to an LBM scheme for Navier Stokes Equations and an Finite Volume scheme for Linearized Euler Equations. The derivation uses the analytical Maxwellian in a BGK model. In this way, we are able to obtain second-order schemes. In addition, we perform an $L^2$-stability analysis. Numerical results validate the approach.

Thomas Camminady, Martin Frank

Sweeping is a commonly used procedure to explicitly solve the discrete ordinates equation, which itself is a common approximation of the neutron transport equation. To sweep through the computational domain, an ordering of the spatial cells is required that obeys the flow of information. We show that this ordering can always be found, assuming a discretization of the spatial domain with regular triangles with no hanging nodes.

Steffen Schotthöfer, M. Paul Laiu, Martin Frank et al.

The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the context of regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy closure as a two-stage approximation to the original entropy closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy.

Jonas Kusch, Graham W. Alldredge, Martin Frank

Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots of specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.

Jonas Kusch, Ryan G. McClarren, Martin Frank

Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the gPC coefficients of the Stochastic Galerkin (SG) approximation, which allows a numerically cheap reduction of oscillations. The derived filter is based on Lasso regression which sets small gPC coefficients of high order to zero. We adaptively choose the filter strength to obtain a zero-valued highest order moment, which allows optimality of the corresponding optimization problem. The filtered SG method is tested for Burgers' and the Euler equations. Results show a reduction of oscillations at shocks, which leads to an improved approximation of expectation values and the variance compared to SG and IPM.

Prince Chidyagwai, Martin Frank, Florian Schneider et al.

The $M_1$ minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the $M_1$ model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.

Richard C. Barnard, Martin Frank, Kai Krycki

In this paper we study the sensitivities of electron dose calculations with respect to the stopping power and the transport coefficients. We focus on the application to radiotherapy simulations. We use a Fokker-Planck approximation to the Boltzmann transport equation. Equations for the sensitivities are derived by the adjoint method. The Fokker-Planck equation and its adjoint are solved numerically in slab geometry using the spherical harmonics expansion ($P_N$) and an HLL finite volume method. Our method is verified by comparison to finite difference approximations of the sensitivities. Finally, we present numerical results of the sensitivities for the normalized average dose deposition depth with respect to the stopping power and transport coefficients, demonstrating the increasing relative sensitivities as beam energy decreases.