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Theory and Modern Applications

Table 2 Accuracy test for equation ( 5.1 )

From: A discontinuous Galerkin finite element method for the Zakharov-Kuznetsov equation

 

N  ×  N

\(\boldsymbol{L^{2}}\) -error

Order

\(\boldsymbol{L^{1}}\) -error

Order

\(Q^{2}\)

10 × 10

4.681719705684662E-002

-

0.938563113113571

-

20 × 20

8.590188004653063E-003

2.45

0.149004060977931

2.66

30 × 30

2.801917125628173E-003

2.76

4.693715377978912E-002

2.85

40 × 40

1.226912433407160E-003

2.87

2.020192920070046E-002

2.93

50 × 50

6.382984939905879E-004

2.93

1.042627582635964E-002

2.96

\(Q^{3}\)

10 × 10

1.951379082541397E-002

-

0.376702045788276

-

20 × 20

2.041593075668467E-003

3.26

3.563208065971012E-002

3.40

30 × 30

4.512659070942643E-004

3.72

7.677136067626609E-003

3.79

40 × 40

1.481541022305029E-004

3.87

2.498721760169104E-003

3.90

50 × 50

6.281947408186389E-005

3.85

1.051975293917244E-003

3.88

  1. Periodic boundary condition in both x- and y-directions in [−30,30]×[−30,30]. Uniform meshes with N × N cells at final time T = 0.0001 and θ = π/6.