Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws

2013 ◽  
Vol 58 (1) ◽  
pp. 41-60 ◽  
Author(s):  
Chao Liang ◽  
Zhengfu Xu
Keyword(s):  
Maximum Principle ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
High Order Schemes ◽  
Flux Limiters

scholarly journals A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

2015 ◽  
Vol 18 (4) ◽  
pp. 901-930 ◽  
Author(s):  
Ziyao Sun ◽  
Honghui Teng ◽  
Feng Xiao
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Solution Structure ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Weno Reconstruction ◽  
Hyperbolic Conservation ◽  
Average Value ◽  
High Order Schemes ◽  
Volume Method

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

scholarly journals Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1885
Author(s):  
Shijian Lin ◽  
Qi Luo ◽  
Hongze Leng ◽  
Junqiang Song
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Euler Method ◽  
High Order ◽  
High Order Accuracy ◽  
Reconstruction Method ◽  
Hyperbolic Conservation ◽  
High Order Schemes ◽  
Numerical Solver ◽  
Polynomial Reconstruction

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem.

scholarly journals High order explicit local time stepping methods for hyperbolic conservation laws

2020 ◽  
Vol 89 (324) ◽  
pp. 1807-1842
Author(s):  
Thi-Thao-Phuong Hoang ◽  
Lili Ju ◽  
Wei Leng ◽  
Zhu Wang
Keyword(s):  
Conservation Laws ◽  
Local Time ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
Time Stepping ◽  
Local Time Stepping
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