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homogenization theory tells us that the macroscale model should be of the form −∇ · (a(x)∇)U(x) = f(x). (2.22) The missing data are the coefficients a0(x). Naturally, for the macroscale solver we… (More)

Singular terms in differential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation… (More)

- Björn Engquist, Yen-Hsi Richard Tsai
- Math. Comput.
- 2005

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different… (More)

- Björn Engquist, Lexing Ying
- SIAM J. Scientific Computing
- 2007

This paper introduces a new directional multilevel algorithm for solving N -body or N -point problems with highly oscillatory kernels. These systems often result from the boundary integral… (More)

- Gil Ariel, Björn Engquist, Yen-Hsi Richard Tsai
- Math. Comput.
- 2009

A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one… (More)

We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for… (More)

- Björn Engquist
- 2011

The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz equation including veryhigh-frequency problems. The first… (More)

Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to… (More)

- Björn Engquist, Lexing Ying
- Multiscale Modeling & Simulation
- 2011

This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure… (More)

Multiscale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of… (More)