# The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

@article{Jin2016TheDS, title={The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients}, author={Shi Jin and Zheng Ma}, journal={Journal of Scientific Computing}, year={2016}, volume={74}, pages={97-121} }

We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singular nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite difference or finite volume approximations in space and time for example, is smoother, we first discretize the equation by a smooth finite difference or…

## 6 Citations

### Stochastic Galerkin-collocation splitting for PDEs with random parameters

- Computer Science, Mathematics
- 2018

For time-dependent, semilinear partial differential equations (PDEs) with random parameters and random initial data, the proposed numerical method is computationally much cheaper than standard stochastic Galerkin methods, and numerical tests show that it outperforms standard Stochastic collocation methods, too.

### A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

- Mathematics, Computer ScienceIMA Journal of Numerical Analysis
- 2019

It turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error, which leads to computable error bounds for the space–stochastic discretization error.

### Uniform Spectral Convergence of the Stochastic Galerkin Method for the Linear Semiconductor Boltzmann Equation with Random Inputs and Diffusive Scalings

- Mathematics, Computer Science
- 2017

The main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

### UNIFORM SPECTRAL CONVERGENCE OF THE STOCHASTIC GALERKIN METHOD FOR THE LINEAR SEMICONDUCTOR BOLTZMANN EQUATION WITH RANDOM INPUTS AND DIFFUSIVE SCALING

- Mathematics, Computer Science
- 2018

The main goal of this paper is to first obtain a sharper estimate on the regularity of the solution–an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

### A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties

- Computer Science, MathematicsArXiv
- 2021

The D2GM is found numerically to be first-order and second-order accurate for (stochastic) linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions, respectively.

### Convergence Analysis on Stochastic Collocation Methods for the Linear Schrodinger Equation with Random Inputs

- Mathematics, Computer ScienceAdvances in Applied Mathematics and Mechanics
- 2020

In this paper, we analyse the stochastic collocation method for a linear Schrödinger equation with random inputs, where the randomness appears in the potential and initial data and is assumed to be…

## References

SHOWING 1-10 OF 28 REFERENCES

### Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2010

### Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials

- MathematicsJ. Sci. Comput.
- 2012

A Galerkin method using bi-orthogonal polynomials is proposed, which decouples the equation in the random spaces, yielding a sequence of uncoupled equations.

### A Stochastic Galerkin Method for Hamilton-Jacobi Equations with Uncertainty

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2015

It is shown that the numerical formulation preserves the symmetry and hyperbolicity of the underlying system, which allows one to efficiently quantify the uncertainty of the Hamilton--Jacobi equations due to random inputs, as demonstrated by the numerical examples.

### Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed

- Computer Science
- 2010

Stochastic collocation methods for the same type of scalar wave equation with random wave speed are investigated and it will be demonstrated that the rate of convergence depends on the regularity of the solutions; and theRegularity is determined by the randomwave speed and the initial and boundary data.

### Galerkin method for wave equations with uncertain coefficients

- Computer Science
- 2008

This paper considers a simple model of a scalar wave equation with random wave speed and shows that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues.

### Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials ∗

- Mathematics
- 2005

When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem…

### High-Order Collocation Methods for Differential Equations with Random Inputs

- Computer ScienceSIAM J. Sci. Comput.
- 2005

A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.

### Approximation results for orthogonal polynomials in Sobolev spaces

- Mathematics
- 1982

We analyze the approximation properties of some interpolation operators and some L2-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight…

### CONVERGENCE ANALYSIS FOR SPECTRAL APPROXIMATION TO A SCALAR TRANSPORT EQUATION WITH A RANDOM WAVE SPEED

- Mathematics, Computer Science
- 2012

This work will prove the spectral convergence of the stochastic Galerkin and collocation methods under some regularity results or assumptions and investigate the errors introduced by discretizations in the random space.