Daniel Appelö

Daniel Appelö, Siyang Wang

We consider wave propagation in a coupled fluid-solid region, separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed, energy based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme.

Adeline Kornelus, Daniel Appelö

A new class of Hermite methods for solving nonlinear conservation laws is presented. While preserving the high order spatial accuracy for smooth solutions in the existing Hermite methods, the new methods come with better stability properties. Artificial viscosity in the form of the entropy viscosity method is added to capture shocks.

Peter DelMastro, Daniel Appelö, Yingda Cheng

We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

Adeline Kornelus, Daniel Appelö

In this work, we outline the entropy viscosity method and discuss how the choice of scaling influences the size of viscosity for a simple shock problem. We present examples to illustrate the performance of the entropy viscosity method under two distinct scalings.