Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes

In this paper, we introduce a new type of troubled-cell indicator to improve hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore, they can reduce computational cost while maintaining non-oscillatory properties in non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid WENO methods. Most of troubled-cell indicators require proper parameters to detect discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are conducted to demonstrate the performance of the proposed indicator. The results with the proposed troubled-cell indicator are in good agreement with pure WENO schemes. Also the new indicator has advantages in the computational cost compared with the other indicators.     

Severe lumbar spinal stenosis combined with Guillain-Barré syndrome: A case report

BACKGROUNDGuillain-Barré syndrome (GBS) is a rare disorder that typically presents with ascending weakness, pain, paraesthesias, and numbness, which mimic the findings in lumbar spinal stenosis. Here, we report a case of severe lumbar spinal stenosis combined with GBS.CASE SUMMARYA 70-year-old man with a history of lumbar spinal stenosis presented to our emergency department with severe lower back pain and lower extremity numbness. Magnetic resonance imaging confirmed the diagnosis of severe lumbar spinal stenosis. However, his symptoms did not improve postoperatively and he developed dysphagia and upper extremity numbness. An electromyogram was performed. Based on his symptoms, physical examination, and electromyogram, he was diagnosed with GBS. After 5 d of intravenous immunoglobulin (0.4 g/kg/d for 5 d) therapy, he gained 4/5 of strength in his upper and lower extremities and denied paraesthesias. He had regained 5/5 of strength in his extremities when he was discharged and had no symptoms during follow-up.CONCLUSIONGBS should be considered in the differential diagnosis of spinal disorder, even though magnetic resonance imaging shows severe lumbar spinal stenosis. This case highlights the importance of a careful diagnosis when a patient has a history of a disease and comes to the hospital with the same or similar symptoms.