Abstract
Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.
Highlights
We consider the two-dimensional general form of the nonlinear hyperbolic partial differential equation (PDE)ut + f (u)x + g(u)y = 0, (1)where u is the unknown function, f is the flux function in x spatial direction, and g is the flux function in y spatial direction
Numerical examples are presented to demonstrate the stability, accuracy, and large CFL numbers for the Weighted essentially nonoscillatory (WENO) spatial discretizations coupled with the third-order Krylov strong stability preserving (SSP) integrating factor Runge–Kutta method for 1D and 2D scalar hyperbolic equations
We developed a class of efficient high-order numerical methods by extending the work in [38] to solve general nonlinear 2D hyperbolic equations
Summary
We consider the two-dimensional general form of the nonlinear hyperbolic partial differential equation (PDE). In [17], sparse grid WENO methods were developed to solve high-dimensional hyperbolic PDEs. Since WENO schemes were designed to handle problems with both complicated solution structures and discontinuities, they require more operations than many other schemes because of their sophisticated nonlinear properties and high-order accuracy. If the high-order spatial method (for example, a WENO scheme) coupled with forward Euler satisfies the nonlinear stability, the same spatial discretization coupled with these higher-order time discretizaton methods will preserve the nonlinear stability Exponential integrators such as integrating factor methods are time schemes which have large linear stability regions. We extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear multi-dimensional problems.
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