Abstract
The main purpose of this paper is to describe a space-time discontinuous Galerin (DG) method based on an extended space-time approximation space for the linear first order hyperbolic equation that contains a high frequency component. We extend the space-time DG spaces of tensor-product of polynomials by adding trigonometric functions in space and time that capture the oscillatory behavior of the solution. We construct the method by combining the basic framework of the space-time DG method with the extended finite element method. The basic principle of the method is integrating the features of the partial differential equation with the standard space-time spaces in the approximation. We present error analysis of the space-time DG method for the linear first order hyperbolic problems. We show that the new space-time DG approximation has an improvement in the convergence compared to the space-time DG schemes with tensor-product polynomials. Numerical examples verify the theoretical findings and demonstrate the effects of the proposed method.
Highlights
In computational acoustics, the medium frequency regime and multiscale wave propagation governed by the wave equation have been gained a constant interest in last decades
The main purpose of this paper is to describe a space-time discontinuous Galerkin (DG) method based on an extended space-time approximation space for the linear ...rst order hyperbolic equation that contains a high frequency component
We presented a space-time discontinuous Galerkin method for the scalar hyperbolic problems that contain high frequency components
Summary
The medium frequency regime and multiscale wave propagation governed by the wave equation have been gained a constant interest in last decades. When multiscale wave propagation presents a high frequency component, developing an e¢ cient numerical methods for these classes of problem is a challenging task. Some example of high frequency problems include the high-intensity focused ultrasound (HIFU) treatment of cancer [1], coupled atomistic continuum modeling in nanomaterials [2] and tunneling in quantum mechanics [3]. The reason for ine¢ ciency of the existing methods is that the standard numerical methods such as the ...nite element (FEM) or discontinuous Galerkin (DG) methods based on semi-discrete approach require a very ...ne mesh in the discretization. Discontinuous Galerkin ...nite element methods, space-time discontinuous Galerkin methods, hyperbolic problems, high frequency solutions
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