smooth initial data
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2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Juntao Huang ◽  
Zhiting Ma ◽  
Yizhou Zhou ◽  
Wen-An Yong

Abstract In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the conservation-dissipation formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod’s shock tube problem although it is trained only with smooth initial data.


2019 ◽  
Vol 12 (3) ◽  
pp. 1215-1230
Author(s):  
Muhammad Asif Gondal ◽  
Inayatur Rehman ◽  
Asima Razzaque

The paper presents error bounds for the second order exponential Runge-Kutta method for parabolic abstract linear time-dependent differential equations incorporating non-smooth initial data. As an example for this particular type of problems, the paper presents a spatial discretization of a partial integro-differential equation arising in financial mathematics, where non-smooth initial conditions occur in option pricing models. For this example, numerical studies of the convergence rate are given


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